The status and potential development of magma power

Energy Vol. 17. No. 6, pp. 547-574, 1992 Printed in Great Britain. All rights reserved

036O-5442/92

$5.W + 0.00 Press plc

Copyright 0 lW2 Pergamon

THE STATUS AND POTENTIAL DEVELOPMENT MAGMA POWER JOHN

149 Eleventh

Street, (Received

Abstract-Since 1975, researchers at magma power development through (DOE). In this paper, we review their the cost of magma power and examine

OF

HARADEN Del Mar, CA 92014. 6 A ugw

U.S.A

1991)

Sandia National Laboratories have investigated the support of the U.S. Department of Energy research. Based on that research, we then estimate its potential development.

INTRODUCTION

Crustal

magma

usually

large

heat can electricity.

bodies

are pockets

and have

of molten

high melting

points,

rock in the crust of the Earth. they contain

large

amounts

Since the bodies of heat

energy.

are The

boil water and the resulting steam can drive conventional turbines to generate The proposed design for generating the electricity consists of drilling a well into the

magma chamber and completing the well with concentric tubing. Cold water then flows down the well annulus, pushes through the magma, warms, returns through the well center, and drives a surface power-plant. Although the concept is simple, it has not yet been implemented. Many engineering Since problems

problems

must still be solved

1975, researchers at under the sponsorship

before

Sandia National of DOE. These

magma

to all the major problems and anticipate no insurmountable solutions. With additional funding from the California Energy are drilling

an exploratory

well near the magma

is a practical

Laboratories have been researchers have developed

source working tentative

of power. on these solutions

problems in implementing the Commission, Sandia researchers

body at Long Valley,

California.

Although

the

well will not approach the magma chamber, the well will validate geophysical measurements, delineate the magma body, and test high-temperature drilling techniques. If the drilling schedule does not slip, the well will be completed in 1992 and tested through 1994. A decision may then be made on drilling the first well to broach a magma chamber. This latter well may remove most of the uncertainty surrounding magma power and may validate or repudiate the proposed technology. If that technology is successful, Valley magma body as a source of electricity. In this paper, examine several

plans

may be made

to develop

the Long

we examine the size and origin of the Long Valley magma body. We then aspects of exploiting the magma body as a source of electricity: the physical the existing research, the proposed technology, the technical problems,

and engineering uncertainties, and the needed

research.

Next,

we examine

the potential

development

of magma

power. We first estimate the cost of magma generated power. Based on that cost, we estimate the market penetration of the new power source. Finally, we present a cost-benefit analysis for drilling the first deep magma well and resolving the uncertainty.

RESOURCE

BASE

state is a silicate melt or Magma is the molten form of igneous rocks. The molten steam, and other gases. There are high-temperature solution of silicates containing water, several different types of magma and they are usually differentiated by their silica contents.‘,2 The polar types of magma are basalt and rhyolite. Both are aluminum silicates. The basalt is richer in iron and manganese. The higher concentration of heavy metals in basalt gives it a higher temperature (near 1250°C). The rhyolite has a temperature near 900°C. 547

548

JOHN HARADEN

Under the western U.S. vast magma deposits lie within 10 km of the surface. The U.S. Geological Survey estimates the heat of those bodies to be between 5.28 X lo’* and 5.28 x 10z3J,” the energy equivalents of 8610 billion and 86,100 billion barrels of oil. By comparison, the vast proven reserves of Saudi Arabia are only 168 billion barrels.4 If all of the magma energy could be converted to electricity, it could supply the U.S. 1980 electrical energy consumption for 6000-60,000 yr.4 If only 25% of the heat in the magma could be recovered, it would exceed the thermal energy of the proven coal reserves of the U.S.4 Some individual magma bodies Nuclear and fossil plants have capacities near could easily drive 1 GW power-plants.” 1000 MW. These numbers only relate to the total energy contained in the magma. Although they describe the size of the resource, they overestimate the usable energy and do not provide a realistic estimate of the development potential. The numbers do not reflect accessibility, recoverability, and conversion efficiency. To cite one extreme example, it seems highly unlikely that a viable power-plant can operate in the crater of Kilauea and can produce uninterrupted electricity. Any effort to tap the magma source of an active volcano would probably require situating the plant some distance from the volcano and slant drilling to the underlying magma body. The resulting cost of drilling might then be exorbitant and might make the cost of power uncompetitive. Many magma bodies may be at great depths, at long distances from populated regions, and at inaccessible locations. They may be impractical to exploit or uneconomical to develop. With the current knowledge about these bodies, it is difficult to estimate the fraction that may be exploitable. However, there is at least one magma body in Long Valley, California, which has been extensively studied and appears to be quite accessible. There is a reliable estimate for the resource size and a tentative system for extracting the energy.3X’ There are probably many other bodies that are just as exploitable, but they have not received as much scrutiny and do not seem as promising. Even if all other magma bodies are impractical, the potential power from Long Valley alone is very significant. Based on the resource size and the tentative extraction system, the reservoir can produce 1.53 X 10zOJ of electricity, which equals the electrical energy generated by burning 75 billion barrels of oil. Assuming an optimistic recovery rate of 40%, Long Valley alone represents the equivalent of 188 billion barrels of oil in place, about the same as the proven reserves of Saudi Arabia. If the conversion efficiency at Long Valley is applied to all magma bodies in the U.S. at depths
Magma power development tectonic

action

will

resupply

the

chamber

from

below

with

549 fresh

magma.

The fresh magma is molten mantle that enters the chamber through bands that are controlled by regional and extensional stresses.’ The tectonic action and the replenishment are usually indicated by earthquakes or other seismic action. This geological setting is important since it increases the energy available at Long Valley. Since the chamber is resupplied, new magma energy would replace some of the energy that had already been extracted. The large surface area of the chamber allows large field development with a 0.04 km* well spacing.’ Many wells could be drilled without solidifying the magma between the wells. Too many wells would halt the convective heat flow and thereby terminate the energy extraction.

MAGMATIC

ORIGIN

Magma bodies, earthquakes, and volcanic activity are not isolated developments. They are manifestations of the same underlying physical forces. The modern theory of plate tectonics provides a uniform explanation for them.%” The theory also explains myriad other geological events. It details the creation of mountain ranges, the generation of ore deposits, and the movements of continents. The theory of plate tectonics divides the outer regions of the Earth into two layers: the lithosphere and the asthenosphere. The lithosphere is the rigid outer layer that is 70-150 km thick. Below the lithosphere is the asthenosphere. It is a plastic region that becomes hotter and more viscous with depth. Although the layer does contain some molten material, the asthenosphere is primarily a ductile solid that flows under stress. The layer resembles glass at room temperature or ice. The glass flows slowly at room temperature. Ice seems brittle; but it flows, as a glacier, down a mountain valley. The lithosphere is divided into about a dozen plates. Below them, the asthenosphere and the rest of the mantle are in plastic flow. Generated by the internal heat of the Earth, vast convection currents cause warmer portions of the plastic mantle to rise and cooler portions to fall. These convection currents cause the plates to move and to move differentially from each other. At the boundaries between two plates, great forces generate earthquakes, volcanic acitivity, and shallow magma bodies. Almost all the volcanic activity and magma creation occur at the plate boundaries. As the plates separate, the downward pressure on the mantle rock diminishes. The mantle then rises, adiabatically decompresses, and melts. The melt collects in a magma chamber at the base of the crust. The magma then flows out of the chamber through a system of vertical passages, fills the gap between the separating plates, cascades down the slope of the ridge and hardens. ,4s the process continues, the cumulative magma flow creates new lithospheric plate material by creating the large ridges or undersea mountain ranges and by producing new sea floor. Since the size of the Earth is essentially constant, new lithosphere can only be created if an equal amount is destroyed. The site of this destruction is the collision of two plates and the formation of a subduction zone, where one plate dives under the other and forms new mantle. Usually an oceanic plate bends and slides under a thicker and more stable continental plate. The line of initial subduction is an oceanic trench. At first, the line of descent is low. It then becomes progressively steeper. The subduction carries some ocean floor sediments and some oceanic crust along with the lithosphere into the mantle. The sediments and the crust contain water that reduces the melting temperature of some components of the subducted material. It also reduces the melting temperature of some components of the mantle material in the overlying plate. As the subducted material plunges downward, the increased heat causes partial melting or fractional distillation of the material. The melted material rises into the overlying rock. There it may cool, crystalize. and form igneous rock masses. It may reach the surface as lava or as explosive

550

JOHN HARADEN

volcanic products like ash and pumice. It may also form shallow magma bodies in the crust of the Earth. The Long Valley magma body is slightly east of the San Andreas fault. That fault is a transform boundary between the Pacific and North American plates. On that boundary, the two plates slide. The movement does not produce separation or subduction, but it does create a geologically active area east of the fault. The area is the Basin and Range Province and includes most of the western U.S. As a result of the plate movement, the area is spreading. The spreading produces regional and extensional forces near Long Valley.1g These create a conduit that allows magma to leak from mantle sources and form the magma body.7 The magma body is primarily rhyolite. The precise genesis of the rhyolite is unknown. This has enormous importance for energy policy. Any knowledge gleaned from the magma body at Long Valley can be applied to other rhyolitic magma bodies. The knowledge is less transferable to other magma types. There may be almost no transfer to basaltic magma bodies. They will probably require a more advanced technology to handle a harsher environment produced by higher temperatures and different magmatic gases. EXISTING

EXPLORATORY

WORK

In 1975, Sandia National Laboratories began an initial 7-yr program to explore the scientific feasibility of magma-energy extraction.2” The program targeted five research areas: source location and definition, source tapping, magma characterization, material compatibility, and energy extraction. Analytical and experimental studies examined all areas. These studies culminated in successful field experiments at Kilauea Iki Lava Lake, Hawaii. Holes were drilled into the molten lava zone, and energy hardware was inserted.21 Two energy extraction systems were tested: a closed system that confines the water working fluid to the well piping, and an open system that allows magma-fluid contact. Thermal experiments demonstrated the possibility of efficiently extracting thermal energy from the direct interaction of water and hot basaltic magma in a melt zone.22-25 Those experiments also estimated probable energyextraction rates from a magma well. All results indicated the scientific feasibility of energy extraction from active magma bodies. This conclusion was backed by the final report of the program and two independent review panels.20*26 In 1984, Sandia began a subsequent program to investigate the engineering feasibility of extracting magma energy from shallow magma bodiesz7 The eventual goal of the program is to drill a deep magma well, install energy-extraction equipment, conduct long-term experiments, and evaluate the economics of magma-energy extraction. The original goal was to complete the program by 1991, but funding difficulties have delayed the program. Potential drilling sites have been assessed for geophysical properties, accessibility, and environmental disruption.‘~~” Seismic studies have mapped the most promising site at Long Valley, California.“-34 Detailed modeling of energy extraction has been coupled with initial estimates of magma-reservoir performance. 3s336A preliminary report on technical and economic feasibility showed the closed system uneconomical and shifted attention to the open system.’ Initial cost modeling indicated the potential economic viability of the open system.s,37*38Subsequently, an advanced numerical code was developed to model more accurately the energy extraction process.“Y,10 More recent cost estimates were then developed.4’34’ In 1988, Sandia obtained permits for a 4-yr exploratory drilling program at Long Valley.41-46 The drilling plan details a well to a depth of 6098 m or to a temperature of 5OOC, whichever is reached first. No effort will be made to penetrate the magma chamber. The well will only validate geophysical measurements and confirm a shallow magma body. The we11 will be completed in 1992 and tested through 1994. A decision will then be made on drilling the first deep magma well. That decision will be based on the potential of magma as an energy source and on the budgetary priorities of the Federal Government.

Magma power development DETAILED

DESCRIPTION

OF THE

ENERGY

551 EXTRACTION

PROCESS

Standard drilling techniques should take the well to the magma chamber. Then the normal bits are replaced with special diamond bits equipped with high-pressure jets. The drilling fluid not only cools the bit and removes the chips, but also chills and solidifies the molten rock ahead of the bit. The diamond studs grind the recently cooled rock. The drilling fluid continues to cool and solidify the rock around the recently made hole. Conventional drilling proceeds, A stabilized hole is created in a solidified rock mass floating in molten rock.*l In contrast to the rate of crustal penetration which varies between 30 and 53 m/day, the probable magma penetration rate is 3 m/day.5,47 Experts at Sandia believe the solidified rock extends to a radius of 10 m from the wellbore, but doubt that the newly created rock has structural integrity. Vertical and horizontal fractures lace the rock. Fractures are frequent, regular, and continuous.8 They pulverize the rock into 5-10 cm cubes.8 The rubble acts as a fractured reservoir. Beyond the rubble is a plastic region that does not support fracturing. 26 It contains the working fluid, prevents its dispersal into the chamber, and forces the fluid back up the well. Water flows through the rubble as oil flows through a cracked sandstone. These predictions are not simply speculative, but are based on extensive experience borrowed from the petroleum industry. Modern reservoir engineering readily explains flow in the solid rind around the wellbore. The well is completed with concentric tubing. The inner and outer tubings are insulated from each other to the magma ceiling. The water flows down the well annulus and returns through the inner tubing. This path allows the water to absorb heat from the geothermal heat gradient of the Earth before the water reaches the magma ceiling. Returning to the surface, the water is insulated from the downward flowing water and the Earth. Then the returning water loses little of its heat and carries almost all of its energy to the surface. In an early design,’ the bottom 15.2 m of the outer pipe are perforated. The annulus is sealed at the top of this perforated section. The inner pipe does not extend below the seal. The next higher 61 m of the outer pipe are not perforated. The remainder of the wellbore is perforated to the magma ceiling. This construction allows water to enter the rubble through the upper perforation, flow down the rubble past the solid section, reenter the lower perforation, and return to the surface through the well center. The current Sandia design specifies a different completion in the magma and involves a crossover at the magma ceiling. The outer tubing stops at the magma ceiling and the inner tubing extends into the magma chamber. The crossover transfers the downward flowing water from the annulus to the inner tubing and allows the water to reach the bottom of the well. The water leaves the tubing at the well bottom, flows back up the fractured rock to the outer annulus at the magma ceiling, passes through the crossover, and returns to the surface in the inner tubing.4X The current Sandia design seems preferable. It is simpler and easier to install. It also provides a better flow path for the water through the fractured zone. As the cold water touches the magma, heat is transferred from the magma to the water. The water temperature rises; the magma temperature falls. With more magma-fluid contact, there is more heat transfer and energy extraction. A fluid-flow rate is calculated to maintain a temperature differential of 200-300°C between the fluid and the magma. The fluid and rubble maintain an equilibrium temperature between 600 and 700°C. Under these conditions. Sandia experts estimate a wellbore heat extraction rate of 0.125 MW/m.’ At the surface, supercritical steam spews from the well. The hot working fluid drives a conventional steam power-plant. After being throttled at constant enthalpy, the steam passes through a turbine and discharges at sub-atmospheric pressure. A condenser strips the remaining enthalpy from the fluid, and a pump returns the fluid to the well. The net power of the plant is the gross power of the turbine minus the power needed for the feedwater and

552

JOHN HARADEN

cooling-water pumps. For a 1 km magma penetration, Sandia analysts expect the net power of the plant to be 50 MW.8 If the working fluid dissolves non-condensible gases as it moves through the rubble, there will be a small problem with the surface power-plant. After passing through the turbine and the condenser, the gases will not condense and flow down the well. A reboiler can rectify the problem. After leaving the well, the hot gases pass through the reboiler, the steam cools into water, the gases are separated from the water, and the water is reheated into steam. The problem can also be solved by redesigning the plant as a binary unit. The working fluid then leaves the well, passes through a heat exchanger, and returns down the well. By pressurizing the fluid, the gases never leave solution. A second working fluid passes through the other side of the heat exchanger, absorbs the heat, and cycles through the turbine. The binary plant is more expensive and less efficient. Small quantities of non-condensible gases do not justify the expense. There must be a serious pollution or corrosion problem, but neither seems likely. The plastic zone isolates the magma reservoir from the solidified and fractured magma that hugs the wellbore. Since the plastic zone is under great pressure, it probably deforms and loses all porosity. The plastic zone then resembles an impermeable clay and prevents fluids from moving from the magma reservoir to the fractured flow area. If there is no fluid communication through the plastic zone, no fresh contaminants can enter the fractured flow area. Within the solidified rock that forms the flow area, the contaminants are limited. They can be removed and the flow area can be cleaned by initially running the plant and discharging the effluent. After scouring the flow area, a new working fluid can be recycled. Any remaining dissolved minerals or entrained solids can be filtered. TECHNICAL

UNCERTAINTIES

Despite the extensive research on magma, there is much uncertainty about the feasibility of magma-energy extraction. Many doubts remain about the nature of magma, about the ability of harnessing its energy, and about the economics of generating electricity from it. Since no deep well has broached a magma chamber, there is no precise characterization of the magma regime. There are only inferences from experimentation and research. Even the exact size and placement of the magma body are unknown. Although the chamber is certainly hot and corrosive, there is no delineation of its composition. Temperature and heat-extraction rates are problematical. Since there is no precise knowledge about drilling in a magma regime, actual drilling may be difficult or impossible. The proposed drilling technology relies heavily on drilling practices in the petroleum and geothermal industries. This drilling technology is effective until subsurface temperatures reach 400°C. Beyond that temperature, drilling becomes speculative.5 The subsurface temperature profile is probably not linear. If the transition zone between 400 and 900°C occurs over a short distance, current technology may only be pressed for a short period and may reach the magma chamber with little difficulty. When the magma chamber is broached, the unknown nature of the magma regime provides potentially serious problems. High temperatures, high formation pressures, and corrosive gases will severely test current technology and may demand new methods and materials. Some answers can be anticipated on the basis of laboratory research; some information can be inferred from the behavior of lava lakes. But these sources are incomplete. Laboratory experiments are based on expectations of the magma regime, and lava lakes are not entirely representative of deeply buried magma. In the lakes, the lava is partially outgassed in the eruption and is under relatively little pressure.@ The crucial test for materials is corrosion at high temperatures. There is a dearth of information about these combined effects in a high-pressure environment. Some of the ignorance stems from incomplete materials testing; the remainder from uncertainty about the composition and concentration of corrosive gases in the magma. Mineral dissolution is another

Magma power development

553

corrosive hazard. Sulfidation is not induced by rhyolitic magma and oxidation should be the major source of corrosion.26 If the plastic region effectively isolates the magma chamber from the flow area, magmatic gases and minerals may not be a corrosive problem. Any corrosion then depends on the temperature and mating of different materials. High temperatures have an adverse effect on many materials. Steel casing and drill pipe lose strength at elevated temperatures. Some of the strength loss may depend on time and produce eventual failure. Corrosion also increases with temperature. In chemical kinetics, reaction rates often double with temperature increases as small as 10°C. Modest temperature rises may therefore increase corrosion levels by several orders of magnitude. Higher temperatures also degrade drill bits through diminished cooling. For roller cone bits, bearings and seals have shorter lives. For drag bits, the diamond studs have shorter lives. Suitable materials probably exist for casing and drill pipe. Exotic casing materials can withstand temperatures as high as 16OO’C.” Many of these materials are ceramics with little toughness and no ductility. Fortunately, less exotic materials may suffice. Extensive testing of metals implies nickel-based super alloys have sufficient strength and chemical resistance to survive the heat and corrosive stress of a magmatic environment rich with volatile components.‘0 High temperatures also reduce the ability to cement casing. Cemented casing is only possible if there is enough time to install the cement before it hardens. With higher temperatures, the cement cures more quickly; and the working time shortens. There is simply less time to work the cement before it hardens. Longer cementing jobs become impossible. The actual working time depends on more than the subsurface temperature profile. The working time depends on the temperature of the cooled wellbore, the temperature capability of the cement, the time for tripping drill pipe and running casing, and the setting depth of the casing. For the Long Valley Caldera, conventional technology probably limits cemented casing to depths where the subsurface temperature is <310”C.“~47 This limitation applies to all cementing jobs requiring more than 150 sacks or anchoring crucial seals. For these jobs, there is not enough time to get the cement in place and to form a good bond. The temperature is too great and the cement hardens too quickly. Cementing is allowed at 400°C for small jobs where there are short pumping times, where there is adequate cooling, and where the cement is only supportive. Cementing may be extended to temperature regimes above 300°C with the use of automated casing and handling equipment. This equipment is relatively new, but it has been demonstrated on large drilling rigs. By shortening the time for moving drill pipe and casing, the equipment may allow more time to install cement. Brookhaven National Laboratory has also developed new formulae that promise usable cement at 500°C. If the cement is effective, it will also allow cementing at higher temperatures and greater depths. Since the magma chamber has a temperature near 9OO”C, a portion of the well will still have uncemented casing. For that uncemented region, wellbore stability becomes vitally important. The greatest impediment to high-temperature drilling may be the absence of adequate well control or the ability to control the flow of formation fluids from the wellbore.5.47 When the bit penetrates a permeable formation where the formation pressure exceeds the hydrostatic pressure of the drilling fluid, the formation fluids displace the drilling fluid from the wellbore. The initial displacement is a kick. If the initial displacement is not stopped, the formation fluids flow uncontrollably and produce a blowout. This is usually the worst disaster that can strike a drilling operation. It may destroy the rig personnel, the rig, and the well. In the Persian Gulf, a blowout on an offshore platform burned uncontrollably for a year. When the blowout was over, the platform was a twisted wreck. Its greatest value was as a navigational hazard. In the magma chamber and the high-temperature regime that precedes it, the drilling fluids may become less dense near the bit and provide little confining pressure in the wellbore. Then there may be no way to contain high-temperature, high-pressure, formation fluids. If they exist in the magma chamber, they may flow to the surface with no restriction. Continued drilling EGY

17:6-C

554

JOHN HARADEN

may be impossible. The only alternative may be a blowout or a well abandonment. Even if high formation pressures do not exist, hole instability may develop. If the hydrostatic pressure of the drilling fluid against the borehole wall is insufficient, the high downhole temperature may make the establishment of a confining mudcake along the borehole wall impossible. Hole instability and a loss of well control stem from inadequate drilling fluids. The current ones are insufficient for confined drilling in a high-temperature regime. The high temperatures decrease the fluid viscosity, degrade the additives, and reduce the fluid effectiveness. Weighted, water-based fluids are ineffective above 250°C. At this temperature, the water molecules have too much mobility and affect the behavior of the clay particles and their interactions with the chemical additives. Above 250°C weighted oil-based fluids are required.5347 If the temperature becomes too high, even oil based fluids may become ineffective. The high temperatures may degrade the oil bases and their additives. Exotic fluids and additives may be necessary to withstand the high temperatures. These may not exist or may be too expensive. Cost and high magma temperatures may limit or prevent their use. The greatest uncertainty is the ability to penetrate the magma deeply. There is little industry experience with magma drilling. Shallow wells have been drilled in the lava beds of Kilauea, but the greatest magma penetration was 89 rn.‘lX51The preliminary report prepared for Sandia favored a magma penetration of 152 m but questioned deeper penetrations.’ Sandia experts are now more optimistic. They believe a 1000 m penetration possible, and recent laboratory evidence appears to support their belief.40,52 The actual magma penetration may depend on the stability of the solidified rock that extends into the magma and hugs the wellbore. Drilling progress may be slowed or halted if hole instability develops above the bit. Stability can only be reestablished by sufficiently cooling those unstable zones by more fluid circulation or injection.5347 When the increased circulation no longer restores stability, the maximum magma penetration will probably be determined. The stability ultimately depends on the heat-extraction rate and the magma currents. A low heat-extraction rate may limit penetration by preventing solidification around the wellbore. Higher heat-extraction rates produce larger rinds of solidified rock around the wellbores and produce more stability. These heat-extraction rates increase with the temperature difference between the magma and the drilling fluid. Strong convective currents in the magma body may present another problem. If they are sufficiently strong, they may shear the solidified wellbore and destroy the penetration. Sustained energy extraction from the magma well depends on the convection pattern in the magma chamber. The magnitude of the convective heat transfer at the boundary of the solidified magma hinges on two factors: (1) the vigor with which fresh hot magma is transported to the boundary and (2) the efficiency of convection and diffusion at the boundary layer bordering the magma. These factors are difficult to quantify. The magma flow in the chamber will probably be turbulent. Convection physics is not well understood in that environment and is complicated by large variations in magma viscosity. The physics at the boundary layer is equally complex and poorly understood. NEEDED

RESEARCH

The immediate need is new drilling technology that can mitigate the effects of high temperature. Improved drilling fluids will increase well control and reduce the chances of a blow out. Higher temperature cements will allow cementing at higher temperatures and provide more hole stability. Improvements in automated pipe and casing handling equipment will reduce tripping times and allow cementing at higher temperatures. Better seals and bearings will produce longer bit lives. The most promising innovation may be insulated drill pipe.53 Sandia research indicates a modest amount of insulation can keep the drilling fluid below 100°C when the bottomhole temperature is 600°C.26 Thermal analyses show, that even deep within the molten body, drilling

Magma power development

55.5

fluid and tubular temperatures can be maintained below 230°C through the insulated drill pipe. The use of the insulated drill pipe may then transform many of the problems of hightemperature drilling into conventional drilling conditions. The cooler drilling fluid provides a greater temperature difference between the drilling fluid and the wellbore wall, The heat extraction increases and lowers the wellbore temperature. More difficult cementing jobs become possible since the cement has a longer time to cure at a lower temperature. The increased regions of cemented casing then provide more hole stability. The insulated drill pipe will also provide a greater temperature difference between the drilling fluid and the magma. The greater temperature difference will increase the heat extraction in the magma chamber, generate a thicker rind of solidified rock around the wellbore, and allow deeper magma penetrations. The cooler drilling fluid and its additives will also undergo less degradation and provide more well control. The cooier fluid also increases bit life. The preliminary designs for the insulated drill pipe are similar to those for double-wall insulated tubing used for steam injection in enhanced oil-recovery methods. The quality of the insulation is not important since a thermal conductivity equal to that of Teflon seems adequate. The major structural and thermal challenges are the pipe joints. They must be strong enough to handle the heavier loads of drill pipe, and they must have enough insulation to inhibit heat transfer. The initial designs have ceramic liners along the inner circumference of the joints. Although the designs exist, they still require considerable analysis and prototype development. Other research seems less important. Materials research indicates adequate alloys already exist for a magmatic environment. The major decision is choosing the material best suited to the chemistry of the reservoir. Samples from the exploratory well may provide this information. Further analysis of energy extraction depends on a better understanding of magmatic properties and processes. Computer simulations and experiments with magma simulants may yield little new information. Significant advances probably depend on full-scale testing in magma.

CO,

PRODUCTION

RATES FOR MAGMA AND THE FOSSIL GENERATION OF ELECTRICITY

FUELS

IN THE

Magma and geothermal reservoirs may both produce steam, but they produce steam with different fluid properties. Geothermal fluids frequently contain large quantities of dissolved CO,.‘““’ The relatively clean fluids of the geysers in northern California contain only 0.5% by weight while the fluids of Monte Amiata in Italy contain 20%.5” Other geothermal reservoirs contain varying amounts, but none is indicative of a magma reservoir. There is only a small igneous contribution of CO2 in geothermal fluids. The main source is groundwater saturated with atmospheric CO,.59 Deep magma bodies contain relatively little. The basaltic magma at Kilauea contains an unusually large amount, which amounts to only 0.65%. The basaltic black smokers on the ocean floor contain O.l%.“’ All rhyolitic magmas contain much less CO!. A typical rhyolite contains only 0.03% .61 The rhyolite at Long Valley contains a similar amount. Although the amount of COz in the reservoir is small, the important statistic is the amount released for each kWh of magma-generated power. A worst case is the atmospheric release of all of the available CO*. For each kg of magma, 0.0003 kg COz are then released. Energy extraction terminates when the magma solidifies and convective heat flow ends, which occurs when the magma cools from 900 to 750°C and turns plastic. For each kg of magma, the energy extracted is E(t) = ~~(900 - 75O)“C. Since the expected energy efficiency is 40% ,’ the generated electricity is E(e) = 0.40cP(900 - 7SO)‘C in kJ/kg. The specific heat is cP = 1.046 kJ/kgC’C.(” Computation yields E(e) = 0.0174 kWh/kg. On an electrical basis, the amount of CO1 released by magma is 0.0172 kg/kWh. The worst case scenario may only be important for the long-term generation of magma power. This is especially true if the CO, is released immediately and not gradually over the life

556

JOHN HARADEN

of the reservoir. If the reservoir is not fully exploited,

the more relevant issue may be the speed at which the CO* leaves the reservoir. This speed is affected by the plastic sheath that separates the magma reservoir from the fractured and solidified magma. The solidified magma hugs the wellbore and forms the heat exchanger through which the working fluid flows. Convective magma currents will rapidly move the reservoir CO:! around the plastic sheath. Since the sheath may resemble an impermeable clay, it may prohibit fluid communication between the heat exchanger and the magma reservoir. The only CO* released may be that originally contained in the heat exchanger. After the release, there may be no more CO2 emissions. If magma becomes a long-term energy source, the worst case scenario provides a convenient bound for the CO* pollution. That bound is an order of magnitude less than that for fossil fuels. If this bound is excessive, all magma-generated CO2 can be eliminated by introducing a binary power-plant with a pressurized working fluid flowing through the well system. There will then be less power and more expense, but there will be no CO2 production. As a basis for comparison, values of (kg CO,)/kWh are needed for coal, oil, and natural gas in the production of electricity. The basis begins with methane. For each kg of completely burned methane, the potentially usable energy is 15.46 kWh/kg.63 Since the burning of 1 kg methane produces 2.75 kg COZ, the complete combustion of 1 kg methane produces (0.179 kg CO,)/kWh. A modern combined cycle power-plant has an efficiency of 45%.4 With that conversion efficiency, the CO;? rate is (0.398 kg CO,)/kWh. Natural gas consists primarily of methane. The exact composition varies with the location and the production history of the reservoir. For this work, the gas standard is a volume average of the gas delivered in the U.S. in 1976: 92.88% methane, 3.91% ethane, 0.62% propane, and 2.59% other non-hydrocarbon gases.64,6” For complete combustion, the usable energy is 9103 kcal/m.64~65 The relation between the released CO;? and the usable energy is (kg CO,)/kWh = 0.181 + [(kcal/m3) X (0.1124) - lOOO.O]X 407 X 1O-5.65 For the standard gas, this value is (0.1819 kg CO,)/kWh. This calculation recognizes that natural gas contains 0.92% CO* by volume that is released when the gas is produced from the reservoir and processed.“5 With a 45% conversion efficiency for a combined cycle power-plant, the CO;? emission rate is (0.404 kg CO,)/kWh. Coal composition varies with location. Different grades have different carbon contents and different heating values. The standard for this work is the United Nations definition of a metric ton (1 mt = 1000 kg) of coal equivalent as 1 mt with a heating value of 7OOOcal/g (8140 kWh/mt). 65 Analysis of 617 bituminous coal samples from the eastern U.S. suggests a carbon content of 70.7% for 7OOOcal/g coa1.65,66The corresponding CO2 production rate for the burning of coal is (0.707)(44/12)[(1000 kg)/(8140 kWh)] = (0.319 kg CO,)/kWh. With an efficiency rate of 35% for a flue-gas-desulfurization plant,67 the emission rate is (0.911 kg CO,)/kWh. An atmospheric fluidized bed plant has an efficiency rate of 36%.67 The corresponding emission rate is (0.866 kg CO,)/kWh. A coal-gasification-combined-cycle plant has the highest efficiency with a rate of 39%.67 Its CO2 emission rate is (0.818 kg CO,)/kWh. For this work, the oil standard is the United Nations definition of a ton of oil equivalent as 1.4543 mt of coal equivalent.65 By that definition, a ton of oil equivalent has a heating value of 1.18 x lo4 kWh/mt. Average crude oil has a carbon content of 85%.65@ For complete combustion of oil, the CO* production rate is (0.85)(44/12)[(1000 kg)/(11,800 kWh)] = (0.2634 kg CO,)/kWh. A conventional boiler has an efficiency of 36%.67 The corresponding COZ emission rate is (0.732 kg CO,)/kWh. With a 45% efficiency rate for a combined cycle plant,67 the CO2 emission rate is (0.585 kg CO,)/kWh. Table 1 summarizes the emission rates for magma and the fossil fuels. There is increasing concern about global warming. At the 1990 economic summit, all the industrialized nations except the U.S. pledged to stabilize CO:! emissions in the next century.m,69 By 2005, Germany pledged to reduce its CO;? production to a level 25% lower than its current production. Great Britain pledged to hold its CO* emissions at their current

Magma power development

557

Table 1. CO, production rates for magma and the fossil fuels. (kg CO$Wh Coal. Flue Gas Desulfwization Coal. Atmospheric Fluidized Bed Coal. Gasification Combined Cycle Oil. Convent~onaJ Boiler Oil. Combined Cycle Natural Gas. Combined Cycle Methane, Combined Cycle Magma

0.9110 0.8860 0.8180 0.7320 0.5850 0.4040 0.3980 0.0172

level. Even Japan pledged to reduce its emissions. Germany also announced plants for a tax on CO2 emissions.“’ If the U.S. takes similar actions, magma power may develop an advantage over fossil-fired power. Since there is no cost effective way to remove CO2 emissions from the burning of fossil fuels,71m73any CO;? tax or mandated reduction in CO2 emissions may increase the cost of fossil-fired power. Since magma power produces almost no C02,42,74 efforts to control global warming will hardly affect the cost of magma power. The potential cost of magma power will then fall relative to that of fossil power. POWER

MODEL

We developed mathematical models for momentum and heat of the working fluid as the fluid passes through the well system, the solidified rock around the wellbore, and the power-plant. We then translated these models into FORTRAN code and computer simulated them. These simulations were used to design the power-plant and to validate the well designs. The input variables were the mass flow rate of the fluid, the initial fluid properties, the magma ceiling, and the dimensions of the well. The solution specified the pressure, temperature, and enthalpy at grid points along the flow path. After designing the power-plant to meet the fluid properties at the well-outlet, the computer code was used to calculate the net power produced by the power-plant. The code was then run for several well depths. The well-inlet conditions were always 0.137 MPa, 67”C, and 279 kJ/kg. Depending on the penetration of the magma and on the mass-flow rate of the working fluid, the well-outlet conditions were 22-50 MPa, 537-593°C and 3023-3370 kJ/kg. For a 1 km magma penetration and for a 41 kg/set flow rate, the net power was 44 MW(e). This last value is very close to the Sandia estimate of 50 MW(e).X Most of that difference can be explained by our ignoring the geothermal heat gradient. This is the natural heat of the Earth that increases with depth. By incorporating it, the power output could have increased by about 20% and could have been much nearer to the Sandia estimate. By ignoring the gradient, we develop more conservative estimates for the energy extraction. We averaged the power variations and made a linear fit to the ordered pairs of depth and power. The points are almost linear and are approximated by P = -312.7 + 44.75d

(1)

as a power vs depth curve with d in km and P in MW. Since d, = 1OOOd- 7000 is the magma penetration in m, P = 0.55 + O.O4475d, (2) is the power by penetration

curve with d, in m and P in MW. FIELD

DEVELOPMENT

Our proposed field development at Long Valley is conservative. Wells only cover that portion of the magma body that is within 7 km of the surface. Most of the body is much deeper and may not be accessible with the well design. In the anticipated economic environment, the deeper portions of the body may not even be economical. There is no calculation for the

JOHN HARADEN

558

occasional replenishment of the magma chamber with fresh magma. Under the development plan, replenishment only lengthens the life of the field. Since that life is already long, the discounted value of the additional revenue may not be large. There is no effort to determine the development path that maximizes profit. With renewable resources, there are often two potential development paths. The first develops the field quickly, exhausts the bulk of the existing resource, and abandons the field. The second path develops the field more slowly, augments the withdrawal with replenishment, and exploits the field forever. The first path may not be feasible since there may be no way to drill enough economical wells and to quickly exhaust all the usable energy. The second path is too uncertain since there is no way to anticipate the amount or the timing of the replenishment. The field is fully developed when the maximum number of producing wells is installed. The number depends on the well spacing and the geometry of the magma body. There is currently no precise delineation of the magma body and no exact calculation of the portion that is within 7 km of the surface. From the available seismic data, we assume that portion forms an annulus on a sphere of radius 6 km. The radii of the annulus are estimated at 1.75 and 3.75 km. The surface area above that region is ~t[(3.75)~ - (1.75)‘3 km2 = 34.45 km*. Since the well spacing is 0.04 km*, the field development allows (34.54 km2)(0.04 km2) = 863 wells. For a 1000 m magma penetration, Eq. (2) shows each well can supply 45.30 MW(e). The entire field can deliver 863(45.30) MW(e) = 3.91 X lo4 MW(e) = 3.91 x 10’ kW(e). Since there is an utilization rate of 0.7 in the average power-plant, the plant can produce (3.91 x lO’kW)(24 h/day)(365 day/yr)(0.7) = 2.40 x 101’ kWh/yr. Since the magma body can supply 4.25 X 1013kWh, full field development can deliver power for (4.25 x 1013kWh)/[2.40 x lO”(kWh/yr)~ = 177 yr. Shorter magma penetrations will produce less power per well, deliver less power from the field, and lengthen the field life. QUANTIFICATION

OF THE

UNCERTAINTY

Most of the physical uncertainty that surrounds magma power stems from the ability to penetrate the magma body and drill the high-temperature environment that precedes it. The magma penetration may represent that uncertainty and its probability density function may quantify the uncertainty. The density function follows from estimates of the potential magma penetration of the well. The minimum estimate is 100 m; the maximum, 1000 m; and the most likely, 500 m.75 The minimum estimate follows from the field work at the lava lakes of Kilauea. The maximum estimate follows from the laboratory work at Sandia. A Bayesian approach interprets these subjective estimates. Presuming a probability density function with a triangular shape to represent the current state of prior beliefs, these estimates translate to the density function: 0

f(dp) =

(5.56 (-4.40 0

X

lo-‘j)d, - 5.56 X lop4 x 10-6)d, + 4.40 x 1o-3

ECONOMIC

if d,< 100, if 100 G d, 6 500, if 500 G d, 6 1000, if d, & 1000.

(3)

MODEL

The cost model estimates the opportunity cost of magma power development. Since there is no other apparent use for the land above the magma resource, landowners receive no royalty. Any royalty would be a pure economic rent and would be a transfer payment from the developers to the landowners. The royalty would not measure any cost to society. Since there are no land costs in the Electric Power Research Institute (EPRI) guidelines,67 the resulting cost for magma power can be compared directly with those for other generating options. To

Magma power development

559

avoid the distortional effects of the tax code and the confusion of accounting practices, there are no taxes. There is no uncontroversial choice for the discount rate. Presumably, it should be the real, long-term rate of return on capital. Depending on liquidity and risk, that rate should be somewhere between the Treasury Bill rate and the return on common stocks. The historical averages of these rates between 1926 and 1978 yield a real rate of 0.0% for Treasury Bills and a real rate of 6.4% for common stocks.76 A real return of 4.0% is used in this paper. This choice matches the after-tax rate suggested by EPRI for utilities when planning new power-plants.“’ With this common rate, the cost of magma power can be more accurately compared with the costs of other energy sources. All dollars are 1991 U.S. dollars. When costs are not available in 1991 dollars, their magnitudes and origins are specifically stated. Their values are then immediately adjusted to 1991 dollars by using the consumer price index in the Economic Report of the President.” Well-construction cost Drilling-cost estimates follow from the drilling plan. The plan specifies the length, diameter, and steel type for each casing string; also the rig size, the depths of cemented casing, the wellhead equipment, the blowout equipment, the frequency of bit changes, and the frequencies and types of logging services, as well as the expected drilling time for the whole well. Cost estimates are made for the material and services necessary to complete the well. The cost is calculated for a well with a total dept of 5487 m, with a magma ceiling at 5335 m, and with a magma penetration of 152 m.5V47Incremental drilling and completion-cost functions are then derived for additional crustal and magma penetrations. These functions are added to the cost for the 5487 m well. Their sum is the cost function for drilling and completing a magma well with a variable magma ceiling and with a variable magma penetration. For the 5487 m well, the drilling cost is $2.44 X 10’; the completion cost is $1.018 x 10’; the total well cost is $3.458 x 10’. Tables 2 and 3 present precise breakdowns for these costs. The drilling and completion time for the 5487 m well is estimated at 240 days. For greater crustal and magma penetrations, there are additional rotating charges, drill-string charges, casing costs, cementing costs, completion costs, and contingent costs. The rotating charges included the daily rates for rig, circulating system, fuel, water, drill-string equipment, transportation and supervision. The drill-string charges included the costs for drill bits and damaged drill pipe, drill collars, and drill stabilizers. Normally the drill bits are the only purchased portions of the Table 2. Drilling costs for a 5487 m well U.S. dollars. Intangible Dnlling Costs Road Locatmn. Constructmn. ml Maintcn.uw Drill Rig @ S9030iday tar 203 deyr Move-on/Move-off Fuel. Water. and Transpmm” Drill Slnng Equipment Clrculallo” symn1 Loggtng selvm\ Cenlenting Services Blowout Prevention Equqm~cn~ Miscellaneous Costs and Expendables Supewislon

1991

25o.lxxl I .x30.0(x1 S?O.(XK) 7’)osxw h.Y70,0(X) hJO.(WX) 17ll.~Xil 870,lxw~ IfJO.O(X) l.h’O.O(K) 650,NX) lS.h?O.O(X)

Subtotal Tangible Dnllmg Costs 30” Conductor Casing 20’ 106 Ib/ft K-55 Surface Ca\mg 16” I09 Ib/ft SS-95 Productmn Casmg I 518” 71.8 lb/f{ SS-95 Productwn Llncr I I 518” 7 1.8 lb/f1 HX Produclio” LI”~I 9 5/8” 53.5 Ib/ti HX Productor L lner Casing Hardware Wellhead

I

Subtotal Conrlngency Total Dnllmg

in

IO.o(X) ?3O.(XX1 V7O.wfl 3(XwiXI I(X)SXX~ I5O.txx1 I ?O.lxX) 230.0(X) 4.7 IO.o(K1 4.07o.OrNJ

Cal\

24.400.0(x1

560

JOHN

HARADEN

Table 3. Completion costs for a 5487 m well in 1991 U.S. dollars. Tangible Completion Costs IO 3/4” x 7 5/8” Insulated Production Swine 8 518” x 5” Insulated Production String

Casing Inspection Casing and Tong Services Drill rig @ $9030/day for 37 days Supervision Well Development Subtotal Contingency Total Completion Costs

I.7fM,cw 10,180,ooo

drill-string equipment. The remaining portions are rented. When those rented portions are damaged, they are also purchased. For additional crustal penetration, the rotating charge is $17,28~/day, the drill-string charge is $1269/m, the cost for a 16” casing is $824/m, and the cementing cost is $106/m. The completion cost is $1442/m for insulated production tubing. The contingency is 20%. The crustal penetration rate is assumed to be 41 m/day.38T47 If d, is the magma ceiling in m, the additional cost in dollars for crustal penetration is (1.20)[(17,280/day)/(41 m/day) + (l269/m) 4 (824/m) + (l~/m) f (1442/m)] X (d, - 5335) = 4875(d, - 5335). For additional magma penetration, the rotating charge is $17,2801day, the drill-string charge is $1269/m, and the cost of 95/8” casing is $978/m. The completion cost is $254/m for 5” tubing. The contingent cost is 20%. The magma penetration rate is assumed to be 3 m/day.3R*47 If d, is the total well depth in m, the cost for additional magma penetration is (1.20)[(17,28O/day)/(3 m/day) + (1269/m) + (978/m) + (254/m)] x (d, - d, - 152) = 9916(d, d, - 152). By adding the cost of the 5487 m well and the costs for additional magma and crustal penetrations, the cost in dollars is obtained for a magma well with ceiling d, and total depth d,: cd = 3.458 X 10'+ 4875(d, - 5335) + 9916(d, - d, - 152). Let d, be the magma penetration completion costs becomes

in m. The d, = d, - d,. Since d, = 7000 m, the drilling and cd = 4.12 X lo'+ 9916d,.

(4)

Let cdd be the present value of cd and let r be the real interest rate. Since it takes 230 days to reach the 7000 m magma ceiling and since it takes dJ3 days to penetrate d,, cdd(dp,r) = 4.12 X lo7 exp(-rt,/2)

+ 9916d, exp[-r(t,

+ r,)/2],

(5)

where t, = 2301365 and t2 = (230 + d,/3)/365. Plantsonstruction A reference

cost

plant of 25 MW is designed to handle the expected steam properties from the magma well. The plant is a supercritical steam plant. Table 4 provides a breakdown for its estimated cost of $1640/kW.SV78Economies of scale provide a relation between cost and power that allows cost calculations for comparable plants in a l-50 MW range. For 1976 data on geothermal power-plants, the cost per kW and MW capacity form a linear relation with slope -0.301 of log(cost per kW) on log(~apacity) .58,79That slope and the reference plant [log@), log(1640)] are used to form a linear relation between the log of power and the log of cost. If P is the plant power in MW, and cLw is the cost per kW, the relation is log(c,w) = -0.301 log(P) + 3.636. Eliminating logs, the relation becomes ckw = 4322/P”.301. Then the total cost in dollars cpC for a plant with MW capacity P is cpC= 1000 PckW, or cpC= 4.32 x lo6 P".6w.

Magma

power

development

561

Table 4. Construction costs for a 25 MW power-plant in 1991 U.S. dollars,

Civil and Structural Work Field Cost and Labor

Fluid Collection and Processmg Pipmg Other. MISC. Electrical Work Civil Work Collection Pipe InJection Pipe and Transfer Pumps Control System Budding Field Cost and Labor Engtneering Contingency

64O.OCC 77o.OLxl I oo.ooo 390.000 I .29o.ooo .29O.o00 I .29O,ooO 39O.00 I .29O,alO 2.58O.KKl 3.870,CQO

I

Subtotal

I3,9tnOOO

Total Plant Costs

40.990.KtO

Substituting Eq. (2) for I’, cpc = 4.32 x 106(0.55 + 0.04475d,)0.hyY.

(6)

Annual plant operating and maintenance costs

The plant O&M costs include staff wages and salaries, equipment maintenance, scale removal, cooling-water treatment, clarifer costs, and other miscellaneous costs. The best estimate for these costs would be data from supercritical steam plants in geothermal applications. The working fluid would have a similar temperature and pressure. Unfortunately, data are not available for these plants. Another good source for estimates would be dry-steam plants like those at the Geysers, California, for which data are again not available. Instead, binary plants in geothermal production are chosen as a source for the estimates. These plants do not use working fluids with temperatures and pressures approaching those of a magma plant. These plants may also have more corrosive fluids in their primary cycles than a magma plant. Their maintenance problems may then be more severe and more expensive. Even with these problems, the binary plants may still provide the best estimates. For a 50 MW plant, EPRI recommends $15.6/(kW-yr) and $O.O015/kWh for operating and maintenance costs in 1977 dollars.67 Adjusted for inflation, these costs become $35.3/(kW-yr) and $O.O034/kWh. Using these costs for all plants in a l-50 MW range, the annual operating and maintenance costs become cpO= 56,150P. Substituting Eq, (2) for P, and simplifying. cpO= (3.07 + 0.251dp) x 104.

(7)

These costs may be less indicative for smaller plants. Any resulting error should be small since the operating and maintenance costs are a small part of the cost of electricity. Annual well-maintenance costs

Maintenance costs for geothermal wells depend on specific fluid properties. Industry estimates vary from $20,000 to $100,000 in 1978 dollars for well workovers and acidizing.‘” Office expense for non-manual labor and supplies is estimated at $70,000 in 1977 dollars.” A 10% overhead expense is usually added to the well and field-office expenses. Adjusted for inflation, the high end total is $375,000. Although the magma fluid properties are not expected to be excessively corrosive, we estimate the annual well-maintenance costs at $419,000.” The high-maintenance cost is necessary since the magma well will be substantially deeper than a

JOHN HARADEN

562

geothermal well. Costs for acidizing and well work-overs rise with depth and temperature. Both of these will be substantially greater in a magma well than in a geothermal well. The downtime for the magma well is estimated at 20% and is the standard time for geothermal wells. Project life

Geothermal wells have lives of 5-15 yr.79 The well lives are determined by corrosive fluid properties, by well plugging, and by thermal cooling. None of these seems likely in a magma well. The well life is assumed to be 15 yr. The supercritical steam plant is estimated to have an operating life of 30 yr and to run 70% of that time. These numbers are those recommended by EPRI for binary, geothermal power-plants.‘j’ These numbers may not be precise, but they should be indicative. The construction time for the power-plant is 18 months.5,78 Plant construction begins after the well has been drilled, completed, and tested. This eliminates the possibility of building the plant near a dry or ruined hole. There are two approaches to resolve the discrepancy between the plant life and well life. In the first approach, the life of the power project is the life of the well. The power project terminates with the plugging of the well. For this approach, the plant is movable and has a salvage value equal to the unused time fraction of the plant cost. In the second approach, the project life is the life of the plant. New wells replace the deteriorated ones until the plant life is exhausted. The approach that minimizes cost is used in the model. Busbar cost

With all costs in dollars, let ~(1, i) = drilling costs in year i, ~(2, i) = plant construction costs in year i, ~(3, i) = well-maintenance costs in year i, ~(4, i) = plant maintenance and operating costs in year i, r = the real interest rate, nb(d,) = the number of years required for drilling the well and building the plant, 11w= the well life, s = the salvage value of the plant, and cY= the annual cost that equates the discounted cost and annual cost streams. If the project life equals the well life, the discounting period is nb(d,) + nw, and nb(d,)+nw cy

2

nb(dp)+nw

exp[-r(i

- 0.5)] + s exp{-r[nb(d,)

c

+ nw]} =

i=l

i=nb(d,)+l

4

c

[c(j, i)] exp[-r(i

- 0.5)].

j=l

Solving for cY, nb(dp)+nw C

cy =

i=l

‘, jII,

[c(j,

exp[-r(i

i)]

- 0.5)] -s exp{ -r[nb(d,)

+ nw]}

nb(dp)+nw i=nbg

)+1

P

exP]-r(i

-

WI

If the project life equals the plant life, s = 0, the discounting period is nb(d,) + 30, and nb(dp)+30 C

cy =

i=l

4 jCl

[c(i,

i)l

q[-r(i

-

0.5)]

nb(d,,)+30 c i=nb(d&+l

exp[-r(i

- 0.5)]

is then converted to the cost of electrical power in mills/kWh by transforming the annual cost to mills and by dividing it by the number of kW in a year. The annual cost in mills is cym and is given by cym = 1000~~. The number of kW must reflect the 30% downtime of the plant and 20% shut-in time of the well for maintenance. Since the maintenance is routine and preventive, both the plant and the well are closed simultaneously. The number of kWh in a year is nkwh and is given by n kWh= 0.70(24)(365)(1OOO)P, where 0.70 = the operating fraction of the plant each year, 24 = the number of hours in a day, 365 = the number of days in a year, 1000 = the number of kW in a MW, and P = the plant power in MW. The cost of electrical power is ebb and is computed by ebb = cym/nkwh. cy

Magma power development

Estimated cost of magma-generated

563

power

There is no baseline estimate for the magma penetration. Respected estimates vary between 1.52 and 1000 m.26*5*47Instead of selecting an arbitrary penetration between the extremal estimates, the cost of electrical power is calculated for several penetrations between 50 and 1000 m at depths between 7050 and 8000 m. These electrical costs are displayed in Fig. 1. Cost falls monotonically with penetration. At a penetration of 50 m, the cost is 286 mills/kWh; at a penetration of 1000 m, 41 mills/kWh. By fitting the curve of Fig. 1, chh can be approximated by ebb = 18.82 + 6~45Jd;x”‘“.

(8)

Without breakthroughs in the conversion of solar energy to electricity or a changed political environment for nuclear power, coal may be the most likely competitor to magma in the generation of electricity. Under the same economic assumptions for magma power, the electricity cost for a coal-gasification plant with combined cycling is 44-48 mills/kWh.92 Equation (8) shows that range corresponds to a magma penetration range of 736-876 m. These are deep penetrations that test the proposed drilling technology. But if the magma penetration can exceed 736 m, magma can be a competitive source of electricity. Equation (3) shows there is only a 15% chance of the magma penetration exceeding 736 m. For a new technology, this may not be a relatively low probability of success. The probability is certainly not low for petroleum exploration. With current exploration methods and equipment, only 11% of wildcat

-?? J.? 0

_r-?-?_-_______-__.

200

400 magma

__.__

h(Xf

X00

IO00

penetration (m)

Fig. 1. Electrical cost by magma penetration. Horizontal lines at 44 and 48mills/kWh range for competitive coal costs.

indicate the

JOHN HARADEN

564

wells strike productive reservoirs of oil and gas. The percentage of profitable wells is even smaller. The coal-fired costs may not remain low. Federal legislation may adjust them for the unpaid damages to the environment and to human health. Those adjustments could easily double the current costs of coal-fired power. With that doubling, magma could be competitive with coal at a magma penetration near 220 m and with a probability near 0.96.

POTENTIAL

MARKET

PENETRATION

The first magma well may be drilled as early as 1995. Since it may take 3 yr to drill a well with a 1000 m magma penetration and to build the power-plant, power production cannot begin before 1998. There may be a lengthy testing period, but testing and power production may occur simultaneously. Limited amounts of experimental power may then begin in 1998. This early date for the delivery of experimental power is not farfetched. In 1991, the California Energy Commission awarded $1.5 million for magma research under the assumption that the commercial viability could be established within 5 yr. An independent review panel concluded that all the technical and engineering problems could be solved in 5 yr.83 Some skeptics may still disagree with the early introduction of magma power. They may argue that the development of magma must follow the development of marginal hydrothermal resources and hot dry rock (HDR). This argument seems specious. Marginal hydrothermal resources may never be economical. They may occur at excessive depths and may have inadequate fluid properties. The commercialization of HDR depends on the establishment of an adequate fracture surface between two parallel wellbores in deeply buried rock. This has not yet occurred and may never occur. Magma power promises to bypass these goethermal resources and tap a much hotter heat source. If the proposed technology for magma energy extraction is successful, there may be little reason to develop these other geothermal resources near shallow magma bodies. The potential market for electricity from magma power is baseload generation. The high capital and low operating costs make magma impractical as a source for intermediate and peaking power. The geographical market for magma power from Long Valley probably consists of California, Oregon, and Washington. When selling power in these markets, utilities may not develop the entire field at Long Valley immediately. Conservative utility managers may be reluctant to fully invest in magma power. Their reluctance may not be unreasonable for a speculative technology. Even after the first few wells have been completed, uncertainties may remain. There may be lingering doubts about the longevity of the energy. The technology may not be perfected and may require modifications. These modifications may be disruptive and costly. Rational policy may dictate diversified power sources and a measured development of magma power. Penetration functions may incorporate these factors and may describe a realistic path of field development. The most popular penetration function is the logistic function. It has the form m(t)/n = l/[l + exp(-c

- @)I,

(9)

where m(t) is some measure of the market penetration at time t, n is some measure of the potential market, c is an integration constant, and # is the penetration rate. @ usually depends on the relative profitability of the penetrating innovation. More prifitable innovations have higher values of # and quicker rates of penetration. Logistic penetration functions provide a convenient analytical formulation for modeling the adoption of an innovation. The functions also have a long history of empirical success. Griliches first used them to model the adoption of hybrid corn.84 Mansfield then popularized the functions by applying them to technological diffusion in basic industries.85 Blackman then applied the functions to more industries and more innovations.8”89 Kydes profiled penetration

Magma power development

565

functions for describing market penetration in the energy sector.W In a fashion that may be important for our work, Romeo applied them extensively to the diffusion of machine tools in several industries.“’ Logistic penetration functions may be unusually appropriate for describing the penetration of magma power. Blackman shows logistic penetrations for gas turbines in peaking power and for nuclear reactors in baseload and intermediate power.x6 Peck shows logistic penetrations for 20 smaller innovations in the electric utility industry.“’ Since magma power has many similarities with hydrothermal power, the historical development of the California geysers may indicate the development path of the Long valley magma body. Blair et al show the development of the California geysers follows a logistic path.*” To predict market penetration with the logistic function, values are needed for @ and c. They completely determine the penetration path. Since magma power is an untried technology, these parameters cannot be obtained by fitting the logistic function to historical data. Instead, $ may be interpreted in Mansfield’s theory as a function of the relative cost s and relative profitability n of the innovation. Then @(n, s) may be estimated by forecasting s and n and by substituting them into a fitted version of @(n, s) similar to those of Mansfield and Blackman. Then a planned initial penetration anchors the curve and determines c. Neither Mansfield nor Blackman has an estimated version of $(n, s) for electric utilities. To form an independent estimate, data on the diffusion of generating technology is needed for the electric utilities. Unfortunately, sufficient data could not be found; and no evidence could be found that other researchers had an estimate of $(n, s) for the electric utilities. In the application of penetration functions, the absence of an appropriate estimate of $(n, s) is not unusual. In desperation, other researchers have used an estimate of $(n, s) for a related industry. The coefficients of JG and s are no problem. They are identical in all of Mansfield’s and Blackman’s equations. In Mansfield’s work, it is clearly assumed that these coefficients are uniform across industries and innovations. Mansfield might object to this interpretation; but he would admit that limited data forced him to assume constant coefficients, and that this assumption produced good results.x5 The intercept presents a different problem. Both Mansfield and Blackman have different intercepts for different industries. These intercepts purportedly measure differences among industries. The differences may not be significant. In a slightly different formulation and application, Romeo uses one regression for the diffusion of machine tools across several industries.“’ He effectively averages the intercepts over industries. His results are still good. Using Romeo’s work as a guide, Mansfield’s intercepts are averaged to get the intercept for magma power. Mansfield’s intercepts are averaged instead of Blackman’s since Mansfield’s industries seem more basic and seem more indicative of the electric power industry. The estimated equation for + is then $ = -0.49 + 0.53~ - 0.027s.

(l0)

There may be better solutions to the problem of specifying @(n, s), but none is readily apparent. Let si be the initial investment for the innovation, and let sr be the average assets of the firms that may develop magma power, then s = 100 s;/sr.

(lI) Mansfield does not clearly differentiate between the total cost of the investment and the present value of that investment. Si is assumed to be the total cost, but it is recognized that the assumption may not be important. By Mansfield’s own admission, he only uses approximations for s,.” For magma power, the difference between the investment cost and its present value may not even be large. The real interest rate is only 4%, and the drilling and construction time is <3 yr. Our assumption also provides a closed form for Si that makes exposition easier.

566

JOHN HARADEN

Adding Eqs. (4) and (6), Si = 4.12 X 10’ + 9916d, + 4.32 X 106[0.55 + 0.04475d,]0.699.

(12)

The major electric utilities in California are the firms likely to develop magma power. For Si, the total assets of those firms are averaged.93 The average in dollars is Sf = 14.1 x 109.

(13)

Substituting Eqs. (12) and (13) into Eq. (11) and simplifying yields s = lo-‘[29.3 + 0.00705 d, + 3.07(0.55 + 0.04475 d,)o.699]. Let b, be the payout period for a coal-gasification payout period for a magma plant, then

(14)

plant with combined cycling, and b, be the

JC= b,lb,.

(15)

Electrical sales will repay both investments. For the successful introduction of magma power, there must be a solution that allows magma developers to receive the more expensive coal price for magma power. Higher regulated returns, regulatory lag, or the existence of independent power producers may justify this solution. The appropriate market price is the upper end of the coal-fired cost range. That is the cost at which magma power begins displacing coal-fired power. Under identical economic assumptions for the cost of magma power, that coal cost is 48 mills/kWh.82 From that cost, the operating and fuel costs must be subtracted. They do not repay capital costs; they only cover expenses. The capital cost per kW in dollars for the coal-gasification plant with combined cycling is ckw = 2247.82 The cost includes financing charges for the 3 yr construction period of the plant.94 These charges must be removed for the payout calculation. Let &w be the capital cost per kW without finance charges, then ckw

= +

2 exp[r(i

- 0.5)],

where r is the real interest rate. For T, 0.04 is consistently used. Substituting and solving, ,. ckw = 2113. For the coal plant, 21 mills/kWh repays the capital cost.82 The capacity factor of the plant is 0.70.82 Then each kW of capacity produces 21(8760)(0.70) mills for capital repayment each year. Multiplying and converting to dollars, that financial stream is $128.7. Dividing that number into &w, the payout period in years is 6, = 16.4.

(16)

From Eq. (2), the power of the magma plant for each magma penetration

d, (m) in MW is

P = 0.55 + O.O4475d,. In kW, that power is lOOO(O.55+ O.O4475d,). Since the electrical sales price is the 48 mills/kWh cost of coal power and since the capacity factor of the plant is 0.70, the plant earns 48(1000)(0.55 + 0.04475dJ (8760)(0.70) mills each year. Converting to dollars and simplifying, the yearly income is (1.62 + 0.132dJ X 105. Subtracting the annual well-maintenance costs and the annual O&M costs of the plant from the yearly income and simplifying, (-2.87 + O.l07d,) x lo5 repays the well and plant-capital costs each year. Dividing this amount into Eq. (12) b, = [412 + 0.09916dr f 43.2(0.55 + 0.04475d,)0.699]/(-2.87

+ O.l07d,).

(17)

After calculating b, and b,, n follows easily. Substituting Eqs. (16) and (17) into Eq. (15), n = -(36.4 + 1.36dP)/[319 + O.O7686d, + 33.5(0.55 + 0.04475d,)0.699].

(18)

Magma power development

567

With the previous calculations for s and 3t, $I follows. Substituting Eqs. (14) and (18) into Eq. (10) and simplifying, @= -0.49 - (19.3 - 0.72&,)/[319 - lo-‘[0.791+

+ O.O7686d, + 33.5(0.55 + 0.04475d,)0.699]

O.O002d, + 0.0829(0.55 + 0.04475~$)~.~~].

(19)

Let n(d,) be the power capacity of the fully developed field in MW, and let m(t, dr) be the developed capacity at time t. The initial condition is one well system in 1998. Since the power of each well system is P = 0.55 + O.O4475d, and since the fully developed

field has 863 wells, n(d,) = 863(0.55 + O.O4475d,).

Simplifying, n(d,) = 475 + 38.6d,.

(20)

m(1998, dp) = 0.55 + O.O4475d,.

(21)

For the one well system at t = 1998,

Setting t = 1998 and substituting Eqs. (20) and (21) into Eq. (9), 0.55 + O.O4475d, = (475 + 38.6d,)/[l+

exp(-c

- (p1998)].

Solving, c = ln[(0.55 + O.O4475d,)/(474 + 38.6d,)] - 1998@ Substituting Eq. (20) into Eq. (9), the path of unconstrained capacity becomes m(t, dp) = (475 + 38.6d,)/[l+

exp(-c

market

(22) penetration

- @)I,

in MW (23)

where @= -0.49 - (19.3 - 0.721d,)/[319

+ O.O7686d, + 33.5(0.55 + 0.04475dp)0.6w]

- lo-*[0.791 + O.O002d, + 0.0829(0.55 + 0.04475d,)0.6w], and c = ln[(0.55 + O.O4475d,)/(474 + 38.6d,)] - 1998@ Let q(t, dJ be the corresponding electrical production in kWh from Long Valley at t. Then q(t, dJ = 1000(8760)(0.7)m(t, dp). Multiplying and simplifying, q(t, dp) = [(29.1 + 2.37d,) x lO’]/[l + exp(-c

- @)I.

Power production continues until the magma body solidifies with the extraction 1013kWh. Let t, be the time when solidification occurs. Then t, is the solution to

(24) of 4.25 x

f? I

q(t, dp) dt = 4.25 x 10’“.

1998

Substituting for q(t, d,) and solving, r, = #-’ ln{[exp(c + 1998#) + l] exp(@k,) - l} - @‘c,

(25)

where k, = (4.25 x 10’)/(29.1 + 2.37d,). Market penetration depends on magma penetration. dm(t, d,)/d(d,) measures the dependence. No tractable expression for the derivative could be found. Instead, the derivative was

JOHN HARADEN

I990

2000

2010

2020

2030

2040

2050

year

Fig. 2. The potential production of electricity at Long Valley from wells with 1000 m magma penetration and the level of new and replaceabte electricity demand.

numerically calculated and it was found that dm(t, d,)/d(d,) > 0. This sign shows that deeper magma penetrations provide faster market penetrations. The faster market penetrations stem from the greater profits of the deeper magma penetrations. The greatest proposed magma penetration is 1000 m and produces the fastest market penetration. Figure 2 shows that the potential electrical production from wells with 1OOOm magma penetrations in Long Valley is always less than a common estimate for new and replaceable electricity demand on the West Coast.4-95 Shallower magma penetrations produce slower market penetrations and smaller electrical generations. For all magma penetrations
ANALYSIS

FOR THE FIRST

DEEP

MAGMA

WELL

There is no current generation of electricity. There will be none until the drilling of the first deep magma well. That well will resolve most of the uncertainty that surrounds magma power. The principal sources of the uncertainty are the effectiveness of the proposed technology and the depth to which the magma can be penetrated. This depth is crucial since it determines the

Magma power development

569

power and revenue that can be extracted from the well. The revenue, the well cost, and the O&M costs then determine the cost of magma-generated power. Although this information is important, the first well should only be drilled if the expected benefits exceed the expected costs. The only costs are for drilling the well. Since there are inherent difficulties in doing anything for the first time, the first magma well is probably more expensive than the subsequent ones. The present value of each subsequent well was previously estimated as cdd(dp, r) and assumed that the well is drilled on a routine basis. This may require several years of experience in magma drilling. For hot dry rock wells at 8000 m, the estimated cost of a mature commercial well is one-third that of a burdensome first we11.96Extending this relation to magma drilling, the cost of the first magma well is estimated at 3cdd(dp, r). Since f(d,) is the probability density for the magma penetration, the expected value of the investment cost is

The benefits are the rewards of less expensive power from magma than from coal. In any year, the benefit is the cost reduction multiplied by the amount of magma generated energy. The benefits begin with initial and experimental power production in 1998. They continue until the magma body solidifies at t,. Since the cost of coal-fired power is 48 mills/kWh and since the cost of magma power is c&d,,), the cost reduction per kWh in mills is 4%cbb(dp). The benefit in dollars at t is [48-c&d&] q(t, d&/1000. At the spudding of the well in 1995, the present value of benefits for given penetration d, is [48 - cbb(dp)]q(t, dp) exp[--r(t + 3 - 1998)]/1000

dt,

where r is the discount rate. The yearly benefit is only positive for d, in the interval (736,lOOO). Production will cease for d, < 736. The present value of all expected benefits is b,=

{[48 - cbb(dp)lq(f, dr)

exp[-r(t

+

3 - 1998)]f(d,)/lOOO}

dt d(d,).

(26)

This expression for the benefits assumes that every well must be drilled. Since the first well is drilled for the investment, there are the added benefits of an existing well at a discount rate of 0.04: 1000

b2 =

I 736

ca(dpr O.O4)f(d,) d(d,).

(27)

If the magma penetration of the first well is less than 736m, there are no subsequent wells. Since the cost for the first well is irretrievable, a power-plant may still be built and power may still be produced from the first well. Since there are no subsequent wells, this plant may only run for the 15-yr life of the initial well. Let x(d,) be the unit cost of electricity in mills/kWh that covers the cost of building the power-plant, operating the plant, and maintaining the well. Since P(d,) = 0.55 + O.O4475d,, x(d,)

= 66.3/Z’(d,)“.301

+ (73.4 + 0.408d,)lP(d,).

Then the benefit in mills/kWh from the marginal well is y(d,)

if 48 - x(d,) > 0 if 48-x(d,)CO’

48 - x(d,)

= (o

Let t(r) = exp(0.5 r)[l - exp(-lSr)]l[exp(r) - 11. Th en the expected benefits in U.S. dollars for this marginal well is

value of the discounted

736

b3 = z(r)6132 exp(-1.5

r)

I0

y(d,)P(d,)f(d,)

d(d,).

(28)

570

JOHN HARADEN

‘1

c 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

real interest rate (%)

Fig. 3. Expected net benefit in 1991 U.S. dollars by real interest rate.

Adding Eqs. (26)-(28), loo0

be=

the discounted value of the expected benefits is

I,

I-. L {[48 - cbb(dp)lq(f, &) exp[--r(t

5736

+ 3 - 1998)lf(~,)/I~~

dt d(d,)

JlYYX

1000 +

736 cdd(dp,

O.O4)f(d,) d(d,) + z(r)6132 exp(-1.5

r) I0

I736

Y(d,)W,)f(d,)

d(d,).

Let b,=b,=c,.

Then b, is the present value of the expected net benefit from drilling the magma well. If b, a 0, the magma well is probably profitable. Figure 3 plots the expected net benefit against the real discount rate. As the real discount rate rises from 1 to 15%, the expected net benefit declines monotonically from $4.72 x lo9 to -$1.14 x lo*. The sign of the benefit changes between 7 and 8%. At 7%, the benefit is $3.74 x 16; at 8%, -$4.19 x 10’. Beyond lo%, the benefit barely changes. The intercept on the discount axis corresponds to an internal rate of return near 7.0%. For lower values of the discount rate, the expected net benefit seems quite large.

Magma power development

571

CONCLUSIONS

Magma is a potential source of vast amounts of electricity. Before magma becomes a practical source of power, many engineering problems must still be solved. Through an active research program at Sandia National Laboratories, researchers have addressed these problems and developed tentative solutions. More progress probably depends on large scale tests in magma. Then the drilling of the first deep magma well will validate or repudiate the tentative solutions and determine the feasibility of magma power. This well is only justified by a favorable cost-benefit analysis. The cost-benefit analysis in this paper shows there is incentive for the drilling. There may be more incentive than the analysis shows since the analysis is conservative and may significantly understate the actual rewards. At every crucial juncture, the most adverse approximations have been used in this work. Either costs are overestimated or benefits are underestimated. External costs for the greenhouse emissions and air pollution of fossil fuels are ignored. The technical analysis is equally conservative. The power from each well is probably underestimated, and an optimally sized well is not used. Since the geothermal-heat gradient is ignored in our wellbore-heat-transfer model, the power estimates of this work are about 20% less than those of Sandia workers.8 Instead of the 50 MW(e) estimated for each well with a 1000 m magma penetration by Sandia researchers,8 we use 45.30 MW(e). The reduced power may create less revenue and higher electrical costs in our analysis. Our power vs magma penetration curve is also linear. Recent Sandia research indicates the curve may not be linear.” The curve may be concave downward. Then the power output diminishes less than proportionally with reductions in the magma penetration. More power than occurs at shallower penetrations. The analysis of this work is based around a big-bore well. If a smaller bore is feasible, the same amount of power may still be extracted with lower drilling costs. 5,36 The potential environmental benefits may make magma more appealing as an energy source. If the greenhouse effect is severe and clean coal technologies still dirty the atmosphere, the continued reliance on fossil fuels for electricity generation may then produce large damages for air pollution. Adjusted for these potential damages, the cost of coal-fired power could easily double.97 By increasing the cost of coal-fired power or by decreasing the cost of magma power, magma becomes a more economical source of electricity. With an increase in the relative profitability of magma power, magma becomes competitive with coal at a shorter magma penetration and with a higher probability. The market penetration quickens and produces more power at a lower relative cost. The expected net benefit then increases and may provide an overwhelmingly favorable cost-benefit analysis for magma power.

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