2. Description of Related Art Hydrocarbons obtained from subterranean (e. g. , sedimentary) formations are often used as energy resources, as feedstocks, and as consumer products. Concerns over depletion of available hydrocarbon resources and declining overall quality of produced hydrocarbons have led to development of processes for more efficient recovery, processing and/or use of available hydrocarbon resources. In situ processes may be used to remove hydrocarbon materials from subterranean formations. Chemical and/or physical properties of hydrocarbon material within a subterranean formation may need to be changed to allow hydrocarbon material to be more easily removed from the subterranean formation. The chemical and physical changes. may include in situ reactions that produce removable fluids, composition changes, solubility changes, density changes, phase changes, and/or viscosity changes of the hydrocarbon material within the formation : A fluid may be, but is not limited to, a gas, a liquid, an emulsion, a slurry, and/or a stream of solid particles. that has flow characteristics similar to liquid flow.

A heat source may be used to heat a subterranean formation. Electric heaters may be used to heat the subterranean formation by radiation and/or conduction. An electric heater may resistively heat an element. U. S. Patent No. 2,548, 360 to Germain describes an electric heating element placed within viscous oil within a wellbore. The heater element heats and thins the oil to allow the oil to be pumped from the wellbore. U. S. Patent No. 4,716, 960 to Eastlund et al. describes electrically heating tubing of a petroleum well by passing a relatively low voltage current through the tubing to prevent formation of solids. U. S.

Patent No. 5,065, 818 to Van Egmond describes an electric heating element that is cemented into a well borehole without a casing surrounding the heating element.

U. S. Patent No. 6,023, 554 to Vinegar et al. describes an electric heating element that is positioned within a casing. The heating element generates radiant energy that heats the casing. A granular solid fill material may be placed between the casing and the formation. The casing may conductively heat the fill material, which in turn conductively heats the formation.

U. S. Patent No. 4,570, 715 to Van Meurs et al. describes an electric heating element. The heating element has an electrically conductive core, a surrounding layer of insulating material, and a surrounding metallic sheath. The conductive core may have a relatively low resistance at high temperatures. The insulating material may have electrical resistance, compressive strength, and heat conductivity properties that are relatively high at high temperatures. The insulating layer may inhibit arcing from the core to the

metallic sheath. The metallic sheath may have tensile strength and creep resistance properties that are relatively high at high temperatures.

U. S. Patent No. 5,060, 287 to Van Egmond describes an electrical heating element having a copper- nickel alloy core.

There has been a significant amount of effort to develop methods and systems to economically produce hydrocarbons, hydrogen, and/or other products from hydrocarbon containing formations. At present, however, there are still many hydrocarbon containing formations from which hydrocarbons, hydrogen, and/or other products cannot be economically produced. Thus, there is still a need for improved methods and systems for production of hydrocarbons, hydrogen, and/or other products from various hydrocarbon containing formations.

SUMMARY In an embodiment, an alternating electrical current may be applied to one or more electrical conductors. The electrical conductors may be located in a subsurface or in a subsurface wellbore. The electrical conductors may provide electrically resistive heat output upon application of the alternating electrical current. At least one of the electrical conductors may include electrically resistive ferromagnetic material. The electrically resistive ferromagnetic material may provide heat when alternating current flows through the electrically resistive ferromagnetic material. The electrically resistive ferromagnetic material may provide a reduced amount of heat above or near a selected temperature. In some embodiments, the ferromagnetic material may automatically provide the reduced amount of heat above or near the selected temperature. In certain embodiments, the selected temperature is approximately the Curie temperature of the electrically resistive ferromagnetic, material. In an embodiment, heat may be allowed to transfer from the electrically resistive ferromagnetic material to a part of the subsurface or the subsurface wellbore.

BRIEF DESCRIPTION OF THE DRAWINGS Advantages of the present invention may become apparent to those skilled in the art with the benefit of the following detailed description of the preferred embodiments and upon reference to the accompanying drawings in which: FIG. 1 depicts an illustration of stages of heating a hydrocarbon containing formation.

FIG. 2 shows a schematic view of an embodiment of a portion of an in situ conversion system for treating a hydrocarbon containing formation.

FIG. 3 depicts an embodiment of an insulated conductor heat source.

FIG. 4 depicts an embodiment of a conductor-in-conduit heat source in a formation.

FIGS. 5,6, and 7 depict cross-sectional representations of an embodiment of a temperature limited heater with an outer conductor having a ferromagnetic section and a non-ferromagnetic section.

FIGS. 8,9, 10, and 11 depict cross-sectional representations of an embodiment of a temperature limited heater with an outer conductor having a ferromagnetic section and a non-ferromagnetic section placed inside a sheath.

FIGS. 12,13, and 14 depict cross-sectional representations of an embodiment of a temperature limited heater with a ferromagnetic outer conductor.

FIGS. 15,16, and 17 depict cross-sectional representations of an embodiment of a temperature limited heater with an outer conductor.

FIGS. 18,19, 20, and 21 depict cross-sectional representations of an embodiment of a temperature limited heater.

FIGS. 22,23, and 24 depict cross-sectional representations of an embodiment of a temperature limited heater with an overburden section and a heating section.

FIG. 25 depicts an embodiment of a coupled section of a composite electrical conductor.

FIG. 26 depicts an embodiment of a coupled section of a composite electrical conductor.

FIG. 27 depicts an embodiment of a coupled section of a composite electrical conductor.

FIG. 28 depicts an embodiment of an insulated conductor heater.

FIG. 29 depicts an embodiment of an insulated conductor heater.

FIG. 30 depicts an embodiment of an insulated conductor located inside a conduit.

FIG. 31 depicts an embodiment of a temperature limited heater with a low temperature ferromagnetic outer conductor.

FIG. 32 depicts an embodiment of a conductor-in-conduit temperature limited heater.

FIG. 33 depicts a cross-sectional representation of an embodiment of a conductor-in-conduit temperature limited heater.

FIG. 34 depicts a cross-sectional representation of an embodiment of an insulated conductor-in- conduit temperature limited heater.

FIGS. 35 and 36 depict cross-sectional views of an embodiment of a temperature limited heater that includes an. insulated conductor.

FIGS. 37 and 38 depict cross-sectional views of an embodiment of a temperature limited heater. that includes an insulated conductor.

FIG. 39 depicts an embodiment of a temperature limited heater with current return through the formation.

FIG. 40 depicts a representation of an embodiment of a three-phase temperature limited heater with current connection through the formation.

FIG. 41 depicts an aerial view of the embodiment shown in FIG. 40.

FIG. 42 depicts electrical resistance versus temperature at various applied electrical currents for a 446 stainless steel rod.

FIG. 43 depicts electrical resistance versus temperature at various applied electrical currents for a temperature limited heater.

FIG. 44 depicts power versus temperature at various applied electrical currents for a temperature limited heater.

FIG. 45 depicts electrical resistance versus temperature at various applied electrical currents for a temperature limited heater.

FIG. 46 depicts data for values of skin depth versus temperature for a solid 1"410 stainless steel rod at various applied AC electrical currents.

FIG. 47 depicts temperature versus time for a temperature limited heater.

FIG. 48 depicts temperature versus log time data for a 410 stainless steel rod and a 304 stainless steel rod.

FIG. 49 displays temperature of the center conductor of a conductor-in-conduit heater as a function of formation depth for a Curie temperature heater with a turndown ratio of 2: 1.

FIG. 50 shows corresponding heater heat flux through a formation for a turndown ratio of 2: 1 along with the oil shale richness profile.

FIG. 51 displays heater temperature as a function of formation depth for a turndown ratio of 3: 1.

FIG. 52 shows corresponding heater heat flux through a formation for a turndown ratio of 3: 1 along with the oil shale richness profile.

FIG. 53 shows heater temperature as a function of formation depth for a turndown ratio of 4: 1.

While the invention is susceptible to various modifications and alternative forms, specific embodiments are shown by way of example in the drawings and may be described in detail. The drawings may not be to scale. It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims.

DETAILED DESCRIPTION The following. description generally relates to systems and methods for treating a hydrocarbon containing formation (e. g., a formation containing coal (including lignite, sapropelic coal,, etc.), oil shale, carbonaceous shale, shungites, kerogen, bitumen, oil, kerogen and oil in a low permeability matrix, heavy hydrocarbons, asphaltites, natural mineral waxes, formations wherein kerogen is blocking production of . other hydrocarbons, etc.). Such formations may be treated to yield relatively high quality hydrocarbon products, hydrogen, and/or other products.

"Hydrocarbons"are generally defined as molecules formed primarily by carbon and hydrogen atoms. Hydrocarbons may also include other elements, such as, but not limited to, halogens, metallic elements, nitrogen, oxygen, and/or sulfur. Hydrocarbons may be, but are not limited to, kerogen, bitumen, pyrobitumen, oils, natural mineral waxes, and asphaltites. Hydrocarbons may be located within or adjacent to mineral matrices within the earth. Matrices may include, but are not limited to, sedimentary rock, sands, silicilytes, carbonates, diatomites, and other porous media. "Hydrocarbon fluids"are fluids that include hydrocarbons. Hydrocarbon fluids may include, entrain, or be entrained in non-hydrocarbon fluids (e. g., hydrogen ("H2"), nitrogen ("N2"), carbon monoxide, carbon dioxide, hydrogen sulfide, water, and ammonia).

A"formation"includes one or more hydrocarbon containing layers, one or more non-hydrocarbon layers, an overburden, and/or an underburden. An"overburden"and/or an"underburden"includes one or more different types of impermeable or substantially impermeable materials. For example, overburden and/or underburden may include rock, shale, mudstone, or wet/tight carbonate (i. e. , an impermeable carbonate without hydrocarbons). In some embodiments of in situ conversion processes, an overburden and/or an underburden may include a hydrocarbon containing layer or hydrocarbon containing layers that are relatively impermeable and are not subjected to temperatures during in situ conversion processing that

results in significant characteristic changes of the hydrocarbon containing layers of the overburden and/or underburden. For example, an underburden may contain shale or mudstone. In some cases, the overburden and/or underburden may be somewhat permeable.

The terms"formation fluids"and"produced fluids"refer to fluids removed from a hydrocarbon containing formation and may include pyrolyzation fluid, synthesis gas, mobilized hydrocarbon, and water (steam). The term"mobilized fluid"refers to fluids within the formation that are able to flow because of thermal treatment of the formation. Formation fluids may include hydrocarbon fluids as well as non- hydrocarbon fluids.

A"heat source"is any system for providing heat to at least a portion of a formation substantially by conductive and/or radiative heat transfer.

A"heater"is any system for generating heat in a well or a near wellbore region. Heaters may be, but are not limited to, electric heaters, burners, combustors that react with material in or produced from a formation (e. g. , natural distributed combustors), and/or combinations thereof. A"unit of heat sources" refers to a number of heat sources that form a template that is repeated to create a pattern of heat sources within a formation.

The term"wellbore"refers to a hole in a formation made by drilling or by inserting a conduit into the formation. A wellbore may have a substantially circular cross section, or other cross-sectional shapes (e. g. , circles, ovals, squares, rectangles, triangles, slits, or other regular or irregular shapes). As used herein, the terms"well"and"opening, "when referring to an opening. in the formation, may be used interchangeably with the term"wellbore." "Insulated conductor"refers to any elongated material that is able to conduct electricity and'that is covered, in whole or in part, by an electrically insulating material. The term"self-controls"refers to controlling an output of a heater without external control of any type.

"Pyrolyzation fluids"or"pyrolysis products"refers to fluid produced substantially during pyrolysis of hydrocarbons. Fluid produced by pyrolysis reactions may mix with other fluids in a formation.

The mixture would be considered pyrolyzation fluid or pyrolyzation product. As used herein,"pyrolysis zone"refers to a volume of a formation (e. g. , a relatively permeable formation such as a tar sands formation) that is reacted or reacting to form pyrolyzation fluid.

"Condensable hydrocarbons"are hydrocarbons that condense at 25 °C at one atmosphere absolute pressure. Condensable hydrocarbons may include a mixture of hydrocarbons having carbon numbers greater than 4. "Non-condensable hydrocarbons"are hydrocarbons that do not condense at 25 °C and one atmosphere absolute pressure. Non-condensable hydrocarbons may include hydrocarbons having carbon numbers less than 5.

Hydrocarbons in formations may be treated in various ways to produce many different products.

In certain embodiments, such formations may be treated in stages. FIG. 1 illustrates several stages of heating a hydrocarbon containing formation. FIG. 1 also depicts an example of yield (barrels of oil equivalent per ton) (y axis) of formation fluids from a hydrocarbon containing formation versus temperature (°C) (x axis) of the formation (as the formation is heated at a relatively low rate).

Desorption of methane and vaporization of water occurs during stage 1 heating. Heating of the formation through stage 1 may be performed as quickly as possible. For example, when a hydrocarbon

containing formation is initially heated, hydrocarbons in the formation may desorb adsorbed methane. The desorbed methane may be produced from the formation. If the hydrocarbon containing formation is heated further, water within the hydrocarbon containing formation may be vaporized. Water may occupy, in some hydrocarbon containing formations, between about 10 % and about 50 % of the pore volume in the formation. In other formations, water may occupy larger or smaller portions of the pore volume. Water typically is vaporized in a formation between about 160 °C and about 285 °C for pressures of about 6 bars absolute to 70 bars absolute. In some embodiments, the vaporized water may produce wettability changes in the formation and/or increase formation pressure. The wettability changes and/or increased pressure may affect pyrolysis reactions or other reactions in the formation. In certain embodiments, the vaporized water may be produced from the formation. In other embodiments, the vaporized water may be used for steam extraction and/or distillation in the formation or outside the formation. Removing the water from and increasing the pore volume in the formation may increase the storage space for hydrocarbons within the pore volume.

After stage 1 heating, the formation may be heated further, such that a temperature within the formation reaches (at least) an initial pyrolyzation temperature (e. g. , a temperature at the lower end of the temperature range shown as stage 2). Hydrocarbons within the formation may be pyrolyzed throughout stage 2. A pyrolysis temperature range may vary depending on types of hydrocarbons within the formation.

A pyrolysis temperature range may include temperatures between about 250 °C and about 900 °C. A pyrolysis temperature range for producing desired products. may extend through only a portion of the total pyrolysis temperature range. In some embodiments, a pyrolysis temperature range for producing desired products may include temperatures between about 250 °C and about 400 °C. If a temperature of hydrocarbons in a formation is slowly. raised through a temperature range from about 250 °C to about 400 °C, production of pyrolysis products may be substantially complete when the temperature approaches 400 °C. Heating the hydrocarbon containing formation with a plurality of heat sources may establish thermal gradients around the heat sources that slowly raise the temperature of hydrocarbons in the formation through a pyrolysis temperature range.

In some in situ conversion embodiments, a temperature of the hydrocarbons to be subjected to pyrolysis may not be slowly increased throughout a temperature range from about 250 °C to about 400 °C.

The hydrocarbons in the formation may be heated to a desired temperature (e. g. , about 325°C). Other temperatures may be selected as the desired temperature. Superposition of heat from heat sources may allow the desired temperature to be relatively quickly and efficiently established in the formation. Energy input into the formation from the heat sources may be adjusted to maintain the temperature in the formation substantially at the desired temperature. The hydrocarbons may be maintained substantially at the desired temperature until pyrolysis declines such that production of desired formation fluids from the formation becomes uneconomical.

Formation fluids including pyrolyzation fluids may be produced from the formation. The pyrolyzation fluids may include, but are not limited to, hydrocarbons, hydrogen, carbon dioxide, carbon monoxide, hydrogen sulfide, ammonia, nitrogen, water, and mixtures thereof. As the temperature of the formation increases, the amount of condensable hydrocarbons in the produced formation fluid tends to decrease. At high temperatures, the formation may produce mostly methane and/or hydrogen. If a

hydrocarbon containing formation is heated throughout an entire pyrolysis range, the formation may produce only small amounts of hydrogen towards an upper limit of the pyrolysis range. After all of the available hydrogen is depleted, a minimal amount of fluid production from the formation will typically occur.

After pyrolysis of hydrocarbons, a large amount of carbon and some hydrogen may still be present in the formation. A significant portion of remaining carbon in the formation can be produced from the formation in the form of synthesis gas. Synthesis gas generation may take place during stage 3 heating depicted in FIG. 1. Stage 3 may include heating a hydrocarbon containing formation to a temperature sufficient to allow synthesis gas generation. For example, synthesis gas may be produced within a temperature range from about 400 °C to about 1200 °C. The temperature of the formation when the synthesis gas generating fluid is introduced to the formation may determine the composition of synthesis gas produced within the formation. If a synthesis gas generating fluid is introduced into a formation at a temperature sufficient to allow synthesis gas generation, synthesis gas may be generated within the formation. The generated synthesis gas may be removed from the formation through a production well or production wells. A large volume of synthesis gas may be produced during generation of synthesis gas.

FIG. 2 shows a schematic view of an embodiment of a portion of an in situ conversion system for treating a hydrocarbon containing formation. Heat sources 100 may be placed within at least a portion of the hydrocarbon containing formation. Heat sources 100 may provide heat to at least a portion of a hydrocarbon containing formation. Energy may be supplied to the heat sources 100 through supply lines 102. The supply lines may be structurally different depending on the type of heat source or heat sources. being used to heat. the formation. Supply lines. for. heat sources may transmit electricity for electric heaters, . may transport fuel for combustors, or may transport heat exchange fluid that is circulated within. the formation.

Production wells 104 may be used to remove formation fluid from the formation.

Formation fluid produced from production wells 104 may be transported through collection piping 106 to treatment facilities 108. Formation fluids may also be produced from heat sources 100. For example, fluid may be produced from heat sources 100 to control pressure within the formation adjacent to the heat sources. Fluid produced from heat sources 100 may be transported through tubing or piping to collection piping 106 or the produced fluid may be transported through tubing or piping directly to treatment facilities 108. Treatment facilities 108 may include separation units, reaction units, upgrading units, fuel cells, turbines, storage vessels, and other systems and units for processing produced formation fluids.

An in situ conversion system for treating hydrocarbons may include barrier wells 110. In certain embodiments, barrier wells 110 may include freeze wells. In some embodiments, barriers may be used to inhibit migration of fluids (e. g. , generated fluids and/or groundwater) into and/or out of a portion of a formation undergoing an in situ conversion process. Barriers may include, but are not limited to naturally occurring portions (e. g. , overburden and/or underburden), freeze wells, frozen barrier zones, low temperature barrier zones, grout walls, sulfur wells, dewatering wells, injection wells, a barrier formed by a gel produced in the formation, a barrier formed by precipitation of salts in the formation, a barrier formed by a polymerization reaction in the formation, sheets driven into the formation, or combinations thereof.

As shown in FIG. 2, in addition to heat sources 100, one or more production wells 104 will typically be placed within the portion of the hydrocarbon containing formation. Formation fluids may be produced through production well 104. In some embodiments, production well 104 may include a heat source. The heat source may heat the portions of the formation at or near the production well and allow for vapor phase removal of formation fluids. The need for high temperature pumping of liquids from the production well may be reduced or eliminated. Avoiding or limiting high temperature pumping of liquids may significantly decrease production costs. Providing heating at or through the production well may: (1) inhibit condensation and/or refluxing of production fluid when such production fluid is moving in the production well proximate the overburden, (2) increase heat input into the formation, and/or (3) increase formation permeability at or proximate the production well. In some in situ conversion process embodiments, an amount of heat supplied to production wells is significantly less than an amount of heat applied to heat sources that heat the formation.

An insulated conductor heater may be a heater element of a heat source. In an embodiment of an insulated conductor heater, the insulated conductor heater is a mineral insulated cable or rod. An insulated conductor heater may be placed in an opening in a hydrocarbon containing formation. The insulated conductor heater may be placed in an uncased opening in the hydrocarbon containing formation. Placing the heater in an uncased opening in the hydrocarbon containing formation may allow heat transfer from the heater to the formation by radiation as well as conduction. Using an uncased opening may significantly reduce heater capital cost by eliminating a need for a portion of casing able to withstand high temperature . conditions. In some heater embodiments, an insulated conductor heater may be placed within a casing in ¢ the formation; may be. cemented within the formation; or may be packed in an opening with sand, gravel, or other fill material. The insulated conductor heater may be supported on a support member positioned within the opening. The support member may be a cable, rod, or a conduit (e. g. , a pipe). The support member may be made of a metal, ceramic, inorganic material, or combinations thereof. Portions of a support member may be exposed to formation fluids and heat during use, so the support member may be chemically resistant and thermally resistant.

Ties, spot welds, and/or other types of connectors may be used to couple the insulated conductor heater to the support member at various locations along a length of the insulated conductor heater. The support member may be attached to a wellhead at an upper surface of the formation. In an embodiment of an insulated conductor heater, the insulated conductor heater is designed to have sufficient structural strength so that a support member is not needed. The insulated conductor heater will in many instances have some flexibility to inhibit thermal expansion damage when heated or cooled.

In certain embodiments, insulated conductor heaters may be placed in wellbores without support members and/or centralizers. An insulated conductor heater without support members and/or centralizers may have a suitable combination of temperature and corrosion resistance, creep strength, length, thickness (diameter), and metallurgy that will inhibit failure of the insulated conductor during use.

One or more insulated conductor heaters may be placed within an opening in a formation to form a heater or heaters. Electrical current may be passed through each insulated conductor heater in the opening to heat the formation. Alternatively, electrical current may be passed through selected insulated conductor heaters in an opening. The unused conductors may be backup heaters. Insulated conductor heaters may be

electrically coupled to a power source in any convenient manner. Each end of an insulated conductor heater may be coupled to lead-in cables that pass through a wellhead. Such a configuration typically has a 180° bend (a"hairpin"bend) or turn located near a bottom of the heater. An insulated conductor heater that includes a 180° bend or turn may not require a bottom termination, but the 180° bend or turn may be an electrical and/or structural weakness in the heater. Insulated conductor heaters may be electrically coupled together in series, in parallel, or in series and parallel combinations. In some embodiments of heaters, electrical current may pass into the conductor of an insulated conductor heater and may be returned through the sheath of the insulated conductor heater.

In the embodiment of a heater depicted in FIG. 3, three insulated conductor heaters 112 are electrically coupled in a 3-phase wye configuration to a power supply. No bottom connection may be required for the insulated conductor heaters. Alternatively, all three conductors of the three-phase circuit may be connected together near the bottom of a heater opening. The connection may be made directly at ends of heating sections of the insulated conductor heaters or at ends of cold pins coupled to the heating sections at the bottom of the insulated conductor heaters. The bottom connections may be made with insulator filled and sealed canisters or with epoxy filled canisters. The insulator may be the same composition as the insulator used as the electrical insulation.

The three insulated conductor heaters depicted in FIG. 3 may be coupled to support member 114 using centralizers 116. Alternatively, the three insulated conductor heaters may be strapped directly to the support tube using metal straps. Centralizers 116 may maintain a location or inhibit movement of insulated conductor heaters 112 on support member 114. Centralizers 116 may be made of metal, ceramic, or combinations thereof. The metal may be stainless steel or any-other type of metal able to. withstand a corrosive and hot environment. In some embodiments ; centralizers 116 may be bowed metal strips welded to the support member. at distances less than about 6 m. A ceramic used in centralizer 116 may be, but is not limited to, A1203, MgO, Si3N4, or other insulator. Centralizers 116 may maintain a location of insulated conductor heaters 112 on support member 114 such that movement of insulated conductor heaters is inhibited at operating temperatures of the insulated conductor heaters. Insulated conductor heaters 112 may also be somewhat flexible to withstand expansion of support member 114 during heating.

Support member 114, insulated conductor heater 112, and centralizers 116 may be placed in opening 118 in hydrocarbon layer 120. Insulated conductor heaters 112 may be coupled to bottom conductor junction 122 using cold pin transition conductor 124. Bottom conductor junction 122 may electrically couple insulated conductor heaters 112 to each other. Bottom conductor junction 122 may include materials that are electrically conducting and do not melt at temperatures found in opening 118.

Cold pin transition conductor 124 may be an insulated conductor heater having lower electrical resistance than insulated conductor heater 112.

Lead-in conductor (s) 126 may be coupled to wellhead 128 to provide electrical power to insulated conductor heater 112. Lead-in conductor 126 may be made of a relatively low electrical resistance conductor such that relatively little heat is generated from electrical current passing through lead-in conductor 126. In some embodiments, the lead-in conductor is a rubber or polymer insulated stranded copper wire (s). In some embodiments, the lead-in conductor is a mineral insulated conductor with a copper core. Lead-in conductor 126 may couple to wellhead 128 at surface 130 through a sealing flange located

between overburden 132 and surface 130. The sealing flange may inhibit fluid from escaping from opening 118 to surface 130.

In some embodiments, reinforcing material 134 may secure overburden casing 136 to overburden 132. In an embodiment of a heater, overburden casing is a 3"diameter carbon steel, Schedule 40 pipe.

Reinforcing material 134 may include, for example, Class G or Class H Portland cement mixed with silica flour for improved high temperature performance, slag or silica flour, and/or a mixture thereof (e. g. , about 1.58 grams per cubic centimeter slag/silica flour). In some heater embodiments, reinforcing material 134 extends radially a width of from about 5 cm to about 25 cm. In some embodiments, reinforcing material 134 may extend radially a width of about 10 cm to about 15 cm.

In certain embodiments, one or more conduits may be provided to supply additional components (e. g., nitrogen, carbon dioxide, reducing agents such as gas containing hydrogen, etc. ) to formation openings, to bleed off fluids, and/or to control pressure. Formation pressures tend to be highest near heating sources. Providing pressure control equipment in heaters may be beneficial. In some embodiments, adding a reducing agent proximate the heating source assists in providing a more favorable pyrolysis environment (e. g. , a higher hydrogen partial pressure). Since permeability and porosity tend to increase more quickly proximate the heating source, it is often optimal to add a reducing agent proximate the heating source so that the reducing agent can more easily move into the formation.

Conduit 138, depicted in FIG. 3, may be provided to add gas from gas source 140, through valve 142, and into opening 118. Conduit 138 and valve 144 may be. used at different times to produce fluids, bleed off pressure, and/or control pressure proximate opening 118. It is to be understood that any of the heating'sources described herein may also be equipped with conduits to supply additional components, to bleed off fluids, and/or to control pressure.

As shown in FIG. 3, support member 114 and lead-in conductor 126 may be. coupled to wellhead 128 at surface 130 of the formation. Surface conductor 156 may enclose reinforcing material 134 and couple to wellhead 128. Embodiments of surface conductor 156 may have an outer diameter of about 10.16 cm to about 30.48 cm or, for example, an outer diameter of about 22 cm. Embodiments of surface conductors may extend to depths of approximately 3 m to approximately 515 m into an opening in the formation. Alternatively, the surface conductor may extend to a depth of approximately 9 m into the opening. Electrical current may be supplied from a power source to insulated conductor heater 112 to generate heat.

Heat generated by an insulated conductor heater may heat at least a portion of a hydrocarbon containing formation. In some embodiments, heat may be transferred to the formation substantially by radiation of the generated heat to the formation. Some heat may be transferred by conduction or convection of heat due to gases present in the opening. The opening may be an uncased opening. An uncased opening eliminates cost associated with thermally cementing the heater to the formation, costs associated with a casing, and/or costs of packing a heater within an opening. In addition, heat transfer by radiation is typically more efficient than by conduction, so the heaters may be operated at lower temperatures in an open wellbore. Conductive heat transfer during initial operation of a heater may be enhanced by the addition of a gas in the opening. The gas may be maintained at a pressure up to about 27 bars absolute.

The gas may include, but is not limited to, carbon dioxide, hydrogen, steam, and/or helium. An insulated

conductor heater in an open wellbore may advantageously be free to expand or contract to accommodate thermal expansion and contraction. An insulated conductor heater may advantageously be removable or redeployable from an open wellbore.

FIG. 4 illustrates an embodiment of a conductor-in-conduit heater that may heat a hydrocarbon containing formation. Conductor 146 may be disposed in conduit 138. Conductor 146 may be a rod or conduit of electrically conductive material. Low resistance sections 148 may be present at both ends of conductor 146 to generate less heating in these sections. Low resistance section 148 may be formed by having a greater cross-sectional area of conductor 146 in that section, or the sections may be made of material having less resistance. In certain embodiments, low resistance section 148 includes a low resistance conductor coupled to conductor 146. In some heater embodiments, conductors 146 may be 316H"347H, 304H, or 310H stainless steel rods with diameters of approximately 2 cm. In some heater embodiments, conductors are 316,304, or 310 stainless steel pipes with diameters of approximately 2.5 cm.

Larger or smaller diameters of rods or pipes may be used to achieve desired heating of a formation. The diameter and/or wall thickness of conductor 146 may be varied along a length of the conductor to establish different heating rates at various portions of the conductor.

Conduit 138 may be made of an electrically conductive material. For example, conduit 138 may be a 3"Schedule 40 pipe made of 347H, 316H, 304H, or 310H stainless steel. Conduit 138 may be disposed in opening 118 in hydrocarbon layer 120. Opening 118 has a diameter able to accommodate conduit 138. A diameter of the opening. may be from about 10 cm to about 22 cm. Larger or smaller diameter openings may. be used to'accommodate particular conduits. or designs.

Conductor 146 may be centered in conduit. 138 by centralizer 150. Gentralizer 150 may electrically isolate conductor 146 from conduit 138. Cenkalizer 150 may inhibit lateral movement and. properly locate conductor 146 Within, conduit 138. Centralizer 150 maybe made of a ceramic material or a combination of ceramic and metallic materials. Centralizers 150 may inhibit deformation of conductor 146 in conduit 138. Centralizer 150 may be spaced at intervals between approximately 0.1 m and approximately 3 m along conductor 146.

A second low resistance section 148 of conductor 146 may couple conductor 146 to wellhead 128, as depicted in FIG. 4. Electrical current may be applied to conductor 146 from power cable 152 through low resistance section 148 of conductor 146. Electrical current may pass from conductor 146 through sliding connector 154 to conduit 138. Conduit 138 may be electrically insulated from overburden casing 136 and from wellhead 128 to return electrical current to power cable 152. Heat may be generated in conductor 146 and conduit 138. The generated heat may radiate within conduit 138 and opening 118 to heat at least a portion of hydrocarbon layer 120. As an example, a voltage of about 480 volts and a current of about 549 amps may be supplied to conductor 146 and conduit 138 in a 229 m (750 ft) heated section to generate about 1150 watts/meter of conductor 146 and conduit 138.

Overburden casing 136 may be disposed in overburden 132. Overburden casing 136 may, in some embodiments, be surrounded by materials that inhibit heating of overburden 132. Low resistance section 148 of conductor 146 may be placed in overburden casing 136. Low resistance section 148 of conductor 146 may be made of, for example, copper welded over carbon steel. Low resistance section 148 may have a diameter between about 2 cm to about 5 cm or, for example, a diameter of about 4 cm. Low resistance

section 148 of conductor 146 may be centralized within overburden casing 136 using centralizers 150.

Centralizers 150 may be spaced at intervals of approximately 6 m to approximately 12 m or, for example, approximately 9 m along low resistance section 148 of conductor 146. In a heater embodiment, low resistance section 148 of conductor 146 is coupled to conductor 146 by a weld or welds. In other heater embodiments, low resistance sections may be threaded, threaded and welded, or otherwise coupled to the conductor. Low resistance section 148 may generate little and/or no heat in overburden casing 136.

Packing material 155 may be placed between overburden casing 136 and opening 118. Packing material 155 may inhibit refluxing fluid from flowing from opening 118 to surface 130.

In a heater embodiment, overburden casing 136 is a 3"Schedule 40 carbon steel pipe. In some embodiments, the overburden casing may be cemented in the overburden. Reinforcing material 134 may be a thermally resistant cement such as 40% silica flour mixed with class I Portland cement. Reinforcing material 134 may extend radially a width of about 5 cm to about 25 cm. Reinforcing material 134 may also be made of material designed to inhibit flow of heat into overburden 132. In other heater embodiments, overburden casing 136 may not be cemented into the formation. Having an uncemented overburden casing may facilitate removal of conduit 138 if the need for removal should arise.

Surface conductor 156 may couple to wellhead 128. Surface conductor 156 may have a diameter of about 10 cm to about 30 cm or, in certain embodiments, a diameter of about 22 cm. Electrically insulating sealing flanges may mechanically couple low resistance section 148 of conductor 146 to wellhead 128 and to electrically couple low resistance section 148 to power cable 152. The electrically insulating sealing flanges may couple power cable 152 to wellhead 128.. For example, power cable 1. 52.. may be a copper cable, wire, or other elongated member. Power cable 152 may include any material'having a substantially low resistance. The power cable may be clamped to an end of the low resistance conductor section to make electrical contact.

In an embodiment, heat may be generated in or by conduit 138. About 10% to about 40%, or, for example, about 20%, of the total heat generated by the heater may be generated in or by conduit 138. Both conductor 146 and conduit 138 may be made of stainless steel. Dimensions of conductor 146 and conduit 138 may be chosen such that the conductor will dissipate heat in a range from approximately 650 watts per meter to 1650 watts per meter. Substantially uniform heating of a hydrocarbon containing formation may be provided along a length of conduit 138 greater than about 300 m or, even greater than about 600 m.

Conduit 158 may be provided to add gas from gas source 140, through valve 142, and into opening 118. An opening is provided in reinforcing material 134 to allow gas to pass into opening 118. Conduit 158 and valve 144 may be used at different times to produce fluids, bleed off pressure, and/or control pressure proximate opening 118. It is to be understood that any of the heating sources described herein may also be equipped with conduits to supply additional components, to produce fluids, and/or to control pressure.

Heat may be generated by the conductor-in-conduit heater within an open wellbore. Generated heat may radiatively heat a portion of a hydrocarbon containing formation adjacent to the conductor-in- conduit heater. To a lesser extent, gas conduction adjacent to the conductor-in-conduit heater may heat a portion of the formation. Using an open wellbore completion may reduce casing and packing costs associated with filling the opening with a material to provide conductive heat transfer between the insulated

conductor and the formation. In addition, heat transfer by radiation may be more efficient than heat transfer by conduction in a formation, so the heaters may be operated at lower temperatures using radiative heat transfer. Operating at a lower temperature may extend the life of the heater and/or reduce the cost of material needed to form the heater.

Some embodiments of heaters may include switches (e. g. , fuses and/or thermostats) that turn off power to a heater or portions of a heater when a certain condition is reached in the heater. In certain embodiments, a"temperature limited heater"may be used to provide heat to a hydrocarbon containing formation. A temperature limited heater generally refers to a heater that regulates heat output (e. g. , reduces heat output) above a specified temperature without the use of external controls such as temperature controllers, power regulators, etc. Temperature limited heaters may be AC (alternating current) electrical resistance heaters. Temperature limited heaters may be more reliable than other heaters. Temperature limited heaters may be less apt to break down or fail due to hot spots in the formation. In some embodiments, temperature limited heaters may allow for substantially uniform heating of a formation. In some embodiments, temperature limited heaters may be able to heat a formation more efficiently by operating at a higher average temperature along the entire length of the heater. The temperature limited heater may be operated at the higher average temperature along the entire length of the heater because power to the heater does not have to be reduced to the entire heater (e. g. , along the entire length of the heater), as is the case with typical heaters, if a temperature along any point of the heater exceeds, or is about. to exceed, a maximum operating temperature of the heater. Portions of a temperature limited heater approaching a Curie, temperature of the heater may automatically reduce the heat output in those portions when a limiting temperature of the heater is approached or reached. The heat output may. automatically reduce due to changes in electrical properties (e. g. ; : electrical resistance) of portions of the temperature.. limited heater at or near a selected temperature. The reduced heat output may be a local effect of a portion of the heater that is at or near the selected temperature. Portions of the heater that are below the selected temperature may have a high heat output, while portions of the heater that are at or near the selected temperature may have a reduced heat output. Thus, a larger power may be supplied to the temperature limited heater during a greater portion of a heating process.

In the context of reduced heat output heating systems, apparatus, and methods, the term "automatically"means such systems, apparatus, and methods acting in a certain way without the use of external control (e. g. , external controllers such as a controller with a temperature sensor and a feedback loop). For example, a system including temperature limited heaters may initially provide a first heat output, and then provide a reduced heat output, near, at, or above a Curie temperature of an electrically resistive portion of the heater when the temperature limited heater is energized by an alternating current.

Temperature limited heaters may be in configurations and/or may include materials that provide automatic temperature limiting properties for the heater at certain temperatures. For example, ferromagnetic materials may be used in temperature limited heater embodiments. Ferromagnetic material may self-limit temperature at or near a Curie temperature of the material to provide a reduced heat output at or near the Curie temperature when an alternating current is applied to the material. In certain embodiments, ferromagnetic materials may be coupled with other materials (e. g., non-ferromagnetic materials and/or highly conductive materials) to provide various electrical and/or mechanical properties.

Some parts of a temperature limited heater may have a lower resistance (caused by different geometries and/or by using different ferromagnetic and/or non-ferromagnetic materials) than other parts of the temperature limited heater. Having parts of a temperature limited heater with various materials and/or dimensions may allow for tailoring a desired heat output from each part of the heater. Using ferromagnetic materials in temperature limited heaters may be less expensive and more reliable than using switches in temperature limited heaters.

Curie temperature is the temperature above which a magnetic material (e. g. , ferromagnetic material) loses its magnetic properties. In addition to losing magnetic properties above the Curie temperature, a ferromagnetic material may begin to lose its magnetic properties when an increasing electrical current is passed through the ferromagnetic material.

A heater may include a conductor that operates as a skin effect heater when alternating current is applied to the conductor. The skin effect limits the depth of current penetration into the interior of the conductor. For ferromagnetic materials, the skin effect is dominated by the magnetic permeability of the conductor. The relative magnetic permeability of ferromagnetic materials is typically greater than 1, and may be greater than 10,100, or even 1000. As the temperature of the ferromagnetic material is raised above the Curie temperature and/or as an applied electrical current is increased, the magnetic permeability of the ferromagnetic material decreases substantially and the skin depth expands rapidly (e. g. , as the inverse square root of the magnetic permeability). The reduction in magnetic permeability results in a decrease in the AC resistance of the conductor near, at, or above the Curie temperature and/or as an applied electrical current is increased. When the heater is powered by a substantially constant current source,. portions of the heater that. approachj reach, or are above the Curie temperature may. have reduced, heat : dissipation. Sections of the heater that are not at or near the Curie temperature may be dominated by skin effect heating that allows the heater to have high heat dissipation.

Curie temperature heaters have been used in soldering equipment, heaters for medical applications, and heating elements for ovens (e. g. , pizza ovens). Some of these uses are disclosed in U. S. Patent Nos.

5,579, 575 to Lamome et al.; 5,065, 501 to Henschen et al.; and 5,512, 732 to Yagnik et al. U. S. Patent No.

4,849, 611 to Whitney et al. describes a plurality of discrete, spaced-apart heating units including a reactive component, a resistive heating component, and a temperature responsive component.

An advantage of using a temperature limited heater to heat a hydrocarbon containing formation may be that the conductor can be chosen to have a Curie temperature in a desired range of temperature operation. The desired operating range may allow substantial heat injection into the formation while maintaining the temperature of the heater, and other equipment, below design temperatures (i. e. , below temperatures that will adversely affect properties such as corrosion, creep, and/or deformation). The temperature limiting properties of the heater may inhibit overheating or burnout of the heater adjacent to low thermal conductivity"hot spots"in the formation. In some embodiments, a temperature limited heater may be able to withstand temperatures above about 250 °C, about 500 °C, about 700 °C, about 800 °C, about 900 °C, or higher depending on the materials used in the heater.

A temperature limited heater may allow for more heat injection into a formation than constant wattage heaters because the energy input into the temperature limited heater does not have to be limited to accommodate low thermal conductivity regions adjacent to the heater. For example, in an oil shale

formation in Green River oil shale there is a difference of at least 50 % in the thermal conductivity of the lowest richness oil shale layers (less than about 0.04 L/kg) and the highest richness oil shale layers (greater than about 0.20 L/kg). When heating such a formation, substantially more heat may be transferred to the formation with a temperature limited heater than with a heater that is limited by the temperature at low thermal conductivity layers, which may be only about 0.3 m thick. Because heaters for heating hydrocarbon formations typically have long lengths (e. g. , greater than 10 m, 100 m, or 300 m), the majority of the length of the heater may be operating below the Curie temperature while only a few portions are at or near the Curie temperature of the heater.

The use of temperature limited heaters may allow for efficient transfer of heat to a formation. The efficient transfer of heat may allow for reduction in time needed to heat a formation to a desired temperature. For example, in Green River oil shale, pyrolysis may require about 9.5 to about 10 years of heating when using about a 12 m heater well spacing with conventional constant wattage heaters. For the same heater spacing, temperature limited heaters may allow a larger average heat output while maintaining heater equipment temperatures below equipment design limit temperatures. Pyrolysis in a formation may occur at an earlier time with the larger average heat output provided by temperature limited heaters. For example, in Green River oil shale, pyrolysis may occur in about 5 years using temperature limited heaters with about a 12 m heater well spacing. Temperature limited heaters counteract hot spots due to inaccurate well spacing or drilling where heater wells come to close together.

. Temperature limited heaters may be advantageously used in many other types of hydrocarbon containing formations : For example, in tar sands formations or relatively permeable, formations containing heavy hydrocarbons, temperature limited heaters may be used to provide a controllable low temperature output for reducing the viscosity of fluids at or near the wellbore or in the formation.. Temperature limited. heaters may inhibit excess coke formation due to overheating of the near wellbore region of the formation.

The use of temperature limited heaters may eliminate or reduce the need to perform temperature logging and/or the need to use fixed thermocouples on the heaters to monitor potential overheating at hot spots. The temperature limited heater may eliminate or reduce the need for expensive temperature control circuitry.

A temperature limited heater may be deformation tolerant if localized movement of a wellbore results in lateral stresses on the heater that could deform its shape. Locations along a length of a heater at which the wellbore approaches or closes on the heater may be hot spots where a standard heater overheats and has the potential to burn out. These hot spots may lower the yield strength of the metal, allowing crushing or deformation of the heater. The temperature limited heater may be formed with S curves (or other non-linear shapes) that accommodate deformation of the temperature limited heater without causing failure of the heater.

In some embodiments, temperature limited heaters may be more economical to manufacture or make than standard heaters. Typical ferromagnetic materials include iron, carbon steel, or ferritic stainless steel. Such materials may be inexpensive as compared to nickel-based heating alloys (such as nichrome, Kanthal, etc. ) typically used in insulated conductor heaters. In one embodiment of a temperature limited heater, the heater may be manufactured in continuous lengths as an insulated conductor heater (e. g. , a mineral insulated cable) to lower costs and improve reliability.

In some embodiments, a temperature limited heater may be placed in a heater well using a coiled tubing rig. A heater that can be coiled on a spool may be manufactured by using metal such as ferritic stainless steel (e. g. , 409 stainless steel) that is welded using electrical resistance welding (ERW). To form a heater section, a metal strip from a roll is passed through a first former where it is shaped into a tubular and then longitudinally welded using ERW. The tubular is passed through a second former where a conductive strip (e. g. , a copper strip) is applied, drawn down tightly on the tubular through a die, and longitudinally welded using ERW. A sheath may be formed by longitudinally welding a support material (e. g. , steel such as 347H or 347HH) over the conductive strip material. The support material may be a strip rolled over the conductive strip material. An overburden section of the heater may be formed in a similar manner. In certain embodiments, the overburden section uses a non-ferromagnetic material such as 304 stainless steel or 316 stainless steel instead of a ferromagnetic material. The heater section and overburden section may be coupled together using standard techniques such as butt welding using an orbital welder. In some embodiments, the overburden section material (i. e. , the non-ferromagnetic material) may be pre-welded to the ferromagnetic material before rolling. The pre-welding may eliminate the need for a separate coupling (i. e. , butt welding) step. In an embodiment, a flexible cable (e. g. , a furnace cable such as a MGT 1000 furnace cable) may be pulled through the center after forming the tubular heater. An end bushing on the flexible cable may be welded to the tubular heater to provide an electrical current return path. The tubular heater, including the flexible cable, may be coiled onto a spool before installation into a heater well. In an embodiment, a temperature limited heater may be installed using a coiled tubing rig.

In an embodiment ; a Curie heater includes. a furnace cable inside a ferromagnetic conduit (e. g. , a 3/4"Schedule 80 446 stainless steel pipe).. The ferromagnetic conduit may be clad with copper or another suitable conductive. material. The ferromagnetic conduit may be placed in a deformation-tolerant conduit, or deformation resistant container. The deformation-tolerant conduit may tolerate'longitudinal deformation, radial deformation, and creep. The deformation-tolerant conduit may also support the ferromagnetic conduit and furnace cable. The deformation-tolerant conduit may be selected based on creep and/or corrosion resistance near or at the Curie temperature. In one embodiment, the deformation-tolerant conduit may be 1-1/2"Schedule 80 347H stainless steel pipe (outside diameter of about 4.826 cm) or 1-1/2"Schedule 160 347H stainless steel pipe (outside diameter of about 4.826 cm). The diameter and/or materials of the deformation-tolerant conduit may vary depending on, for example, characteristics of the formation to be heated or desired heat output characteristics of the heater. In certain embodiments, air may be removed from the annulus between the deformation-tolerant conduit and the clad ferromagnetic conduit. The space between the deformation-tolerant conduit and the clad ferromagnetic conduit may be flushed with a pressurized inert gas (e. g. , helium, nitrogen, argon, or mixtures thereof). In some embodiments, the inert gas may include a small amount of hydrogen to act as a"getter"for residual oxygen. The inert gas may pass down the annulus from the surface, enter the inner diameter of the ferromagnetic conduit through a small hole near the bottom of the heater, and flow up inside the ferromagnetic conduit. Removal of the air in the annulus may reduce oxidation of materials in the heater (e. g. , the nickel-coated copper wires of the furnace cable) to provide a longer life heater, especially at elevated temperatures. Thermal conduction between a furnace cable and the ferromagnetic conduit, and between the ferromagnetic conduit and the deformation-tolerant conduit, may be improved when the inert gas is helium. The pressurized inert gas in

the annular space may also provide additional support for the deformation-tolerant conduit against high formation pressures.

Temperature limited heaters may be used for heating hydrocarbon formations including, but not limited to, oil shale formations, coal formations, tar sands formations, and heavy viscous oils. Temperature limited heaters may be used for remediation of contaminated soil. Temperature limited heaters may also be used in the field of environmental remediation to vaporize or destroy soil contaminants. Embodiments of temperature limited heaters may be used to heat fluids in a wellbore or sub-sea pipeline to inhibit deposition of paraffin or various hydrates. In some embodiments, a temperature limited heater may be used for solution mining of a subsurface formation (e. g. , an oil shale or coal formation). In certain embodiments, a fluid (e. g. , molten salt) may be placed in a wellbore and heated with a temperature limited heater to inhibit deformation and/or collapse of the wellbore. In some embodiments, the temperature limited heater may be attached to a sucker rod in the wellbore or be part of the sucker rod itself. In some embodiments, temperature limited heaters may be used to heat a near wellbore region to reduce near wellbore oil viscosity during production of high viscosity crude oils and during transport of high viscosity oils to the surface. In some embodiments, a temperature limited heater may enable gas lifting of a viscous oil by lowering the viscosity of the oil without coking of the oil.

Certain embodiments of temperature limited heaters may be used in chemical or refinery processes at elevated temperatures that require control in a narrow temperature. range to inhibit unwanted chemical reactions or damage from locally elevated temperatures. Some applications may include, but are not limited to, reactor tubes, cokers, and distillation towers. Temperature limited heaters may also be. used in pollution control devices (e. g.,. catalytic converters,. and oxidizers) to. allow rapid heating to a. control , ! temperature without complex temperature control circuitry : Additionally, temperature limited heaters may be used in food processing to avoid damaging food with excessive temperatures. Temperature limited heaters may also be used in the heat treatment of metals (e. g. , annealing of weld joints). Temperature limited heaters may also be used in floor heaters, cauterizers, and/or various other appliances. Temperature limited heaters may be used with biopsy needles to destroy tumors by raising temperatures in vivo.

Some embodiments of temperature limited heaters may be useful in certain types of medical and/or veterinary devices. For example, a temperature limited heater may be used to therapeutically treat tissue in a human or an animal. A temperature limited heater for a medical or veterinary device may have ferromagnetic material including a palladium-copper alloy with a Curie temperature of about 50 °C. A high frequency (e. g. , greater than about 1 MHz) may be used to power a relatively small temperature limited heater for medical and/or veterinary use.

A ferromagnetic alloy used in a Curie temperature heater may determine the Curie temperature of the heater. Curie temperature data for various metals is listed in"American Institute of Physics Handbook," Second Edition, McGraw-Hill, pages 5-170 through 5-176. A ferromagnetic conductor may include one or more of the ferromagnetic elements (iron, cobalt, and nickel) and/or alloys of these elements. In some embodiments, ferromagnetic conductors may include iron-chromium alloys that contain tungsten (e. g., HCM12A and SAVE12 (Sumitomo Metals Co. , Japan) and/or iron alloys that contain chromium (e. g. , Fe- Cr alloys, Fe-Cr-W alloys, Fe-Cr-V alloys, Fe-Cr-Nb alloys). Of the three main ferromagnetic elements, iron has a Curie temperature of about 770 °C ; cobalt has a Curie temperature of about 1131 °C ; and nickel

has a Curie temperature of about 358 °C. An iron-cobalt alloy has a Curie temperature higher than the Curie temperature of iron. For example, an iron alloy with 2% cobalt has a Curie temperature of about 800 °C ; an iron alloy with 12% cobalt has a Curie temperature of about 900 °C ; and an iron alloy with 20% cobalt has a Curie temperature of about 950 °C. An iron-nickel alloy has a Curie temperature lower than the Curie temperature of iron. For example, an iron alloy with 20% nickel has a Curie temperature of about 720 °C, and an iron alloy with 60% nickel has a Curie temperature of about 560 °C.

Some non-ferromagnetic elements used as alloys may raise the Curie temperature of iron. For example, an iron alloy with 5.9% vanadium has a Curie temperature of about 815 °C. Other non- ferromagnetic elements (e. g. , carbon, aluminum, copper, silicon, and/or chromium) may be alloyed with iron or other ferromagnetic materials to lower the Curie temperature. Non-ferromagnetic materials that raise the Curie temperature may be combined with non-ferromagnetic materials that lower the Curie temperature and alloyed with iron or other ferromagnetic materials to produce a material with a desired Curie temperature and other desired physical and/or chemical properties. In some embodiments, the Curie temperature material may be a ferrite such as NiFe204. In other embodiments, the Curie temperature material may be a binary compound such as FeNi3 or Fe3Al.

Magnetic properties generally decay as the Curie temperature is approached. The "Handbook of Electrical Heating for Industry"by C. James Erickson (IEEE Press, 1995) shows a typical curve for 1% carbon steel (i. e. , steel with 1% carbon by weight). The loss of magnetic permeability starts at temperatures above about 650 sand tends to be complete, when temperatures exceed about 730 °C. Thus, .

. the self. limiting temperature may be somewhat below an actual Curie temperature of a ferromagnetic conductor. The skin depth for current flow in. 1% carbon steel is about 0. 132. cm at room temperature and increases to about 0. 445 at about 720 °C. : From about 720 °C. to about 730. °C, the skin depth sharply increases to over 2.5 cm. Thus, a temperature limited heater embodiment using 1% carbon steel may self- limit between about 650 °C and about 730 °C.

Skin depth generally defines an effective penetration depth of alternating current into a conductive material. In general, current density decreases exponentially with distance from an outer surface to a center along a radius of a conductor. The depth at which the current density is approximately 1/e of the surface current density is called the skin depth. For a solid cylindrical rod with a diameter much greater than the penetration depth, or for hollow cylinders with a wall thickness exceeding the penetration depth, the skin depth, 6, is: (1) 6 = 1981.5* ( (p/ (R*f))'/' ; in which: 6 = skin depth in inches; p = resistivity at operating temperature (ohm-cm); u = relative magnetic permeability; and f = frequency (Hz).

EQN. 1 is obtained from the"Handbook of Electrical Heating for Industry"by C. James Erickson (IEEE Press, 1995). For most metals, resistivity (p) increases with temperature. The relative magnetic permeability generally varies with temperature and with current. Additional equations may be used to assess the variance of magnetic permeability and/or skin depth on both temperature and/or current. The dependence of zut on current arises from the dependence of g on the magnetic field.

Materials used in a temperature limited heater may be selected to provide a desired turndown ratio.

Turndown ratio for a temperature limited heater is the ratio of the highest AC resistance just below the Curie temperature to the highest AC resistance just above the Curie temperature. Turndown ratios of at least 2: 1,3 : 1,4 : 1,5 : 1, or greater may be selected for temperature limited heaters. A selected turndown ratio may depend on a number of factors including, but not limited to, the type of formation in which the temperature limited heater is located (e. g. , a higher turndown ratio may be used for an oil shale formation with large variations in thermal conductivity between rich and lean oil shale layers) and/or a temperature limit of materials used in the wellbore (e. g. , temperature limits of heater materials). In some embodiments, a turndown ratio may be increased by coupling additional copper or another good electrical conductor to a ferromagnetic material (e. g. , adding copper to lower the resistance above the Curie temperature).

A temperature limited heater may provide a minimum heat output (i. e. , minimum power output) below the Curie temperature of the heater. In certain embodiments, the minimum heat output may be at least about 400 W/m, about 600 W/m, about 700 W/m, about 800 W/m, or higher. The temperature limited heater may reduce the heat output above the Curie temperature. The reduced heat output is typically substantially less than the heat output below the Curie temperature. In some embodiments, the reduced heat output may be less than about 400 W/m, less than about 200 W/m, or may approach 100 W/m.

In some embodiments, a temperature limited heater may operate substantially independently of the thermal load on the heater in a certain operating temperature range."Thermal load"is the rate that heat is transferred from a heating system to its surroundings. It is to. be understood that the thermal. load may. vary, with temperature of the surroundings and/or the thermal conductivity of the surroundings. In an embodiment, a temperature limited heater may operate at or above a Curie temperature of. the heater such that the operating temperature. of the. heater. does. not-vary by more than about 1. 5 °C for a decrease in. thermal load of about 1 W/m proximate to a portion of the. heater. In some embodiments, the operating temperature of the heater may not vary by more than about 1 °C, or by more than about 0.5 °C for a decrease in thermal load of about 1 W/m.

The AC resistance or heat output of a portion of a temperature limited heater may decrease sharply above the Curie temperature of the portion due to the Curie effect. In certain embodiments, the value of the AC resistance or heat output above or near the Curie temperature is less than about one-half of the value of AC resistance or heat output at a certain point below the Curie temperature. In some embodiments, the heat output above or near the Curie temperature may be less than about 40%, 30%, 20%, 15%, or 10%, of the heat output at a certain point below the Curie temperature (e. g. , about 30 °C below the Curie temperature, about 40 °C below the Curie temperature, about 50 °C below the Curie temperature, or about 100 °C below the Curie temperature). In certain embodiments, the AC resistance above or near the Curie temperature may decrease to about 80%, 70%, 60%, or 50%, of the AC resistance at a certain point below the Curie temperature (e. g. , about 30 °C below the Curie temperature, about 40 °C below the Curie temperature, about 50 °C below the Curie temperature, or about 100 °C below the Curie temperature).

In some embodiments, AC frequency may be adjusted to change the skin depth of a ferromagnetic material. For example, the skin depth of 1% carbon steel at room temperature is about 0.132 cm at 60 Hz, about 0.0762 cm at 180 Hz, and about 0.046 cm at 440 Hz. Since heater diameter is typically larger than twice the skin depth, using a higher frequency (and thus a heater with a smaller diameter) may reduce

equipment costs. For a fixed geometry, a higher frequency results in a higher turndown ratio. The turndown ratio at a higher frequency may be calculated by multiplying the turndown ratio at a lower frequency by the square root of the higher frequency divided by the lower frequency. In some embodiments, a frequency between about 100 Hz and about 600 Hz may be used. In some embodiments, a frequency between about 140 Hz and about 200 Hz may be used. In some embodiments, a frequency between about 400 Hz and about 550 Hz may be used.

To maintain a substantially constant skin depth until the Curie temperature of a heater is reached, the heater may be operated at a lower frequency while the heater is cold and operated at a higher frequency while the heater is hot. Line frequency heating is generally favorable, however, because there is less need for expensive components (e. g. , power supplies that alter frequency). Line frequency is the frequency of a general supply (e. g. , a utility company) of current. Line frequency is typically 60 Hz, but may be 50 Hz or other frequencies depending on the source (e. g. , the geographic location) for the supply of the current.

Higher frequencies may be produced using commercially available equipment (e. g. , solid state variable frequency power supplies). In some embodiments, electrical voltage and/or electrical current may be adjusted to change the skin depth of a ferromagnetic material. Increasing the voltage and/or decreasing the current may decrease the skin depth of a ferromagnetic material. A smaller skin depth may allow a heater with a smaller diameter to be used, thereby reducing equipment costs. In certain embodiments, the applied current may be at least about 1 amp, about 10 amps, about 70 amps, 100 amps, 200 amps, 500 amps, or greater. In some embodiments, alternating current may be supplied at voltages above about 220 volts,. above about 480 volts, above about 600 volts, above about 1000 volts, or above about 1500 volts.

In an embodiment, a temperature limited heater may include an inner conductor inside an outer. conductor.. The inner conductor and the outer conductor-may be radially disposed about a central axis. The inner and outer conductors. may. be separated by an insulation layer. In certain embodiments, the inner and outer conductors may be coupled at the bottom of the heater. Electrical current may flow into the heater through the inner conductor and return through the outer conductor. One or both conductors may include ferromagnetic material.

An insulation layer may comprise an electrically insulating ceramic with high thermal conductivity, such as magnesium oxide, aluminum oxide, silicon dioxide, beryllium oxide, boron nitride, silicon nitride, etc. The insulating layer may be a compacted powder (e. g. , compacted ceramic powder).

Compaction may improve thermal conductivity and provide better insulation resistance. For lower temperature applications, polymer insulation made from, for example, fluoropolymers, polyimides, polyamides, and/or polyethylenes, may be used. The insulating layer may be chosen to be infrared transparent to aid heat transfer from the inner conductor to the outer conductor. In an embodiment, the insulating layer may be transparent quartz sand. The insulation layer may be air or a non-reactive gas such as helium, nitrogen, or sulfur hexafluoride. If the insulation layer is air or a non-reactive gas, there may be insulating spacers designed to inhibit electrical contact between the inner conductor and the outer conductor. The insulating spacers may be made of, for example, high purity aluminum oxide or another thermally conducting, electrically insulating material such as silicon nitride. The insulating spacers may be a fibrous ceramic material such as Nextel 312, mica tape, or glass fiber. Ceramic material may be made of alumina, alumina-silicate, alumina-borosilicate, silicon nitride, or other materials.

An insulation layer may be flexible and/or substantially deformation tolerant. For example, if the insulation layer is a solid or compacted material that substantially fills the space between the inner and outer conductors, the heater may be flexible and/or substantially deformation tolerant. Forces on the outer conductor can be transmitted through the insulation layer to the solid inner conductor, which may resist crushing. Such a heater may be bent, dog-legged, and spiraled without causing the outer conductor and the inner conductor to electrically short to each other. Deformation tolerance may be important if a wellbore is likely to undergo substantial deformation during heating of the formation.

In certain embodiments, the outer conductor may be chosen for corrosion and/or creep resistance. In one embodiment, austentitic (non-ferromagnetic) stainless steels such as 304H, 347H, 347HH, 316H, or 31 OH stainless steels may be used in the outer conductor. The outer conductor may also include a clad conductor. For example, a corrosion resistant alloy such as 800H or 347H stainless steel may be clad for corrosion protection over a ferromagnetic carbon steel tubular. If high temperature strength is not required, the outer conductor may also be constructed from a ferromagnetic metal with good corrosion resistance (e. g. , one of the ferritic stainless steels). In one embodiment, a ferritic alloy of 82.3% iron with 17. 7% chromium (Curie temperature 678 °C) may provide desired corrosion resistance.

The Metals Handbook, vol. 8, page 291 (American Society of Materials (ASM) ) shows a graph of Curie temperature of iron-chromium alloys versus the amount of chromium in the alloys. In some temperature limited heater embodiments, a separate support rod or tubular (made from, e. g. , 347H stainless steel) may be coupled to a heater (e. g. , a heater made from an iron/chromium alloy) to provide. strength and/or creepxesistance. The support material and/or the. ferromagnetic, material may be selected to provide. a 100, 000 hour creep-rupture strength of at least, 3, 000 psi at about 650 °C., In some embodiments, the 100 ; 000 hour creep-rupture strength may be at least about 2,000 psi at about 650 °C or at leastabout 1, 000 : psi at about 650 °C. For example, 347H steel ! has a favorable creep-rupture strength at or above 650°C. In some embodiments, the 100,000 hour creep-rupture strength may range from about 1,000 psi to about 6,000 psi or more for longer heaters and/or higher earth or fluid stresses.

In an embodiment with an inner ferromagnetic conductor and an outer ferromagnetic conductor, the skin effect current path occurs on the outside of the inner conductor and on the inside of the outer conductor. Thus, the outside of the outer conductor may be clad with a corrosion resistant alloy, such as stainless steel, without affecting the skin effect current path on the inside of the outer conductor.

A ferromagnetic conductor with a thickness greater than the skin depth at the Curie temperature may allow a substantial decrease in AC resistance of the ferromagnetic material as the skin depth increases sharply near the Curie temperature. In certain embodiments (e. g. , when not clad with a highly conducting material such as copper), the thickness of the conductor may be about 1.5 times the skin depth near the Curie temperature, about 3 times the skin depth near the Curie temperature, or even about 10 or more times the skin depth near the Curie temperature. If the ferromagnetic conductor is clad with copper, thickness of the ferromagnetic conductor may be substantially the same as the skin depth near the Curie temperature. In some embodiments, a ferromagnetic conductor clad with copper may have a thickness of at least about three-fourths of the skin depth near the Curie temperature.

In an embodiment, a temperature limited heater may include a composite conductor with a ferromagnetic tubular and a non-ferromagnetic, high electrical conductivity core. The non-ferromagnetic,

high electrical conductivity core may reduce a required diameter of the conductor. For example, the conductor may be a composite 1.19 cm diameter conductor with a core of 0.575 cm diameter copper clad with a 0.298 cm thickness of ferritic stainless steel or carbon steel surrounding the core. A composite conductor may allow the electrical resistance of the temperature limited heater to decrease more steeply near the Curie temperature. As the skin depth increases near the Curie temperature to include the copper core, the electrical resistance may decrease more sharply.

A composite conductor may increase the conductivity of a temperature limited heater and/or allow the heater to operate at lower voltages. In an embodiment, a composite conductor may exhibit a relatively flat resistance versus temperature profile. In some embodiments, a temperature limited heater may exhibit a relatively flat resistance versus temperature profile between about 100 °C and about 750 °C, or in a temperature range between about 300 °C and about 600 °C. A relatively flat resistance versus temperature profile may also be exhibited in other temperature ranges by adjusting, for example, materials and/or the configuration of materials in a temperature limited heater.

In certain embodiments, the relative thickness of each material in a composite conductor may be selected to produce a desired resistivity versus temperature profile for a temperature limited heater. In an embodiment, the composite conductor may be an inner conductor surrounded by 0.127 cm thick magnesium oxide powder as an insulator. The outer conductor may be 304H stainless steel with a wall thickness of 0.127 cm. The outside diameter of the heater may be about 1.65 cm.

A composite conductor. (e. g. , a composite inner conductor or a composite outer conductor) may be manufactured by. methods including, but not limited to, coextrusion, roll forming, tight fit tubing (erg., cooling the inner'member and heating the outer member, then inserting the inner member in the outer member, followed by a drawing operation and/or allowing-the system to cool), explosive or electromagnetic cladding, arc overlay welding, longitudinal strip welding, plasma powder welding, billet coextrusion, electroplating, drawing, sputtering, plasma deposition, coextrusion casting, magnetic forming, molten cylinder casting (of inner core material inside the outer or vice versa), insertion followed by welding or high temperature braising, shielded active gas welding (SAG), and/or insertion of an inner pipe in an outer pipe followed by mechanical expansion of the inner pipe by hydroforming or use of a pig to expand and swage the inner pipe against the outer pipe. In some embodiments, a ferromagnetic conductor may be braided over a non-ferromagnetic conductor. In certain embodiments, composite conductors may be formed using methods similar to those used for cladding (e. g. , cladding copper to steel). A metallurgical bond between copper cladding and base ferromagnetic material may be advantageous. Composite conductors produced by a coextrusion process that forms a good metallurgical bond (e. g. , a good bond between copper and 446 stainless steel) may be provided by Anomet Products, Inc. (Shrewsbury, MA).

In an embodiment, two or more conductors may be joined to form a composite conductor by various methods (e. g. , longitudinal strip welding) to provide tight contact between the conducting layers. In certain embodiments, two or more conducting layers and/or insulating layers may be combined to form a composite heater with layers selected such that the coefficient of thermal expansion decreases with each successive layer from the inner layer toward the outer layer. As the temperature of the heater increases, the innermost layer expands to the greatest degree. Each successive outward lying layer expands to a slightly

lesser degree, with the outermost layer expanding the least. This sequential expansion may provide relatively intimate contact between layers for good electrical contact between layers.

In an embodiment, two or more conductors may be drawn together to form a composite conductor.

In certain embodiments, a relatively malleable ferromagnetic conductor (e. g. , iron such as 1018 steel) may be used to form a composite conductor. A relatively soft ferromagnetic conductor typically has a low carbon content. A relatively malleable ferromagnetic conductor may be useful in drawing processes for forming composite conductors and/or other processes that require stretching or bending of the ferromagnetic conductor. In a drawing process, the ferromagnetic conductor may be annealed after one or more steps of the drawing process. The ferromagnetic conductor may be annealed in an inert gas atmosphere to inhibit oxidation of the conductor. In some embodiments, oil may be placed on the ferromagnetic conductor to inhibit oxidation of the conductor during processing.

The diameter of a temperature limited heater may be small enough to inhibit deformation of the heater by a collapsing formation. In certain embodiments, the outside diameter of a temperature limited heater may be less than about 5 cm. In some embodiments, the outside diameter of a temperature limited heater may be less than about 4 cm, less than about 3 cm, or between about 2 cm and about 5 cm.

In heater embodiments described herein (e. g. , including, but not limited to, temperature limited heaters, insulated conductor heaters, conductor-in-conduit heaters, and elongated member heaters), a largest transverse cross-sectional dimension of a heater may be selected to provide a desired ratio of the largest transverse cross-sectional dimension to. wellbore diameter (e. g. , initial wellbore diameter).. The largest transverse cross-sectional dimension is the largest dimension of the heater on the same axis, as the wellbore diameter (e. g., the diameter of a cylindrical heater-or the width of a vertical heater). In certain : embodiments, the ratio of the largest transverse cross-sectional dimension to wellbore diameter may be selected to be less than about 1: 2, less than about 1 : 3, or less than about 1 : 4. The ratio of heater diameter to wellbore diameter may be chosen to inhibit contact and/or deformation of the heater by the formation (i. e., inhibit closing in of the wellbore on the heater) during heating. In certain embodiments, the wellbore diameter may be determined by a diameter of a drill bit used to form the wellbore.

In an embodiment, a wellbore diameter may shrink from an initial value of about 17 cm to about 6 cm during heating of a formation (e. g. , for a wellbore in oil shale with a richness greater than about 0.12 L/kg). At some point, expansion of formation material into the wellbore during heating results in a balancing between the hoop stress of the wellbore and the compressive strength due to thermal expansion of hydrocarbon, or kerogen, rich layers. At this point, the formation may no longer have the strength to deform or collapse a heater, or a liner. For example, the radial stress provided by formation material may be about 12000 psi at a diameter of about 17 cm, while the stress at a diameter of about 6 cm after expansion may be about 3000 psi. A heater diameter may be selected to be less than about 5.1 cm to inhibit contact of the formation and the heater. A temperature limited heater may advantageously provide a higher heat output over a significant portion of the wellbore (e. g. , the heat output needed to provide sufficient heat to pyrolyze hydrocarbons in a hydrocarbon containing formation) than a constant wattage heater for smaller heater diameters (e. g. , less than about 5.1 cm).

In certain embodiments, a heater may be placed in a deformation resistant container. The deformation resistant container may provide additional protection for inhibiting deformation of a heater.

The deformation resistant container may have a higher creep-rupture strength than a heater. In one embodiment, a deformation resistant container may have a creep-rupture strength of at least about 3000 psi at 100,000 hours for a temperature of about 650 °C. In some embodiments, the creep-rupture strength of a deformation resistant container may be at least about 4000 psi at 100,000 hours, or at least about 5000 psi at 100,000 hours for a temperature of about 650 °C. In an embodiment, a deformation resistant container may include one or more alloys that provide mechanical strength. For example, a deformation resistant container may include an alloy of iron, nickel, chromium, manganese, carbon, tantalum, and/or mixtures thereof.

FIG. 5 depicts an embodiment of a temperature limited heater with an outer conductor having a ferromagnetic section and a non-ferromagnetic section. FIGS. 6 and 7 depict cross-sectional views of the embodiment shown in FIG. 5. In one embodiment, ferromagnetic section 160 may be used to provide heat to hydrocarbon layers in the formation. Non-ferromagnetic section 162 may be used in an overburden of the formation. Non-ferromagnetic section 162 may provide little or no heat to the overburden, thus inhibiting heat losses in the overburden and improving heater efficiency. Ferromagnetic section 160 : may include a ferromagnetic material such as 409 or 410 stainless steel. 409 stainless steel may be readily available as strip material. Ferromagnetic section 160 may have a thickness of about 0.3 cm. Non- ferromagnetic section 162 may be copper with a thickness of about 0.3 cm. Inner conductor 164 may be copper. Inner conductor 164 may have a diameter of about 0.9 cm. Electrical insulator 166 may be magnesium oxide powder or other suitable insulator material. Electrical insulator 166 may have a thickness of about 0. 1 cm to about 0. 3 cm.

FIG.. 8 depicts an embodiment of a temperature limited heater with an outer conductor having jazz ferromagnetic section and a non-ferromagnetic section placed inside a sheath.-FIGS. 9,10, and 11 depict cross-sectional views of the embodiment shown in FIG. 8. Ferromagnetic. section 160 may be 410. stainless steel with a thickness of about 0.6 cm. Non-ferromagnetic section 162 may be copper with a thickness of about 0.6 cm. Inner conductor 164 may be copper with a diameter of about 0.9 cm. Outer conductor 168 may include ferromagnetic material. Outer conductor 168 may provide some heat in the overburden section of the heater. Providing some heat in the overburden may inhibit condensation or refluxing of fluids in the overburden. Outer conductor 168 may be 409,410, or 446 stainless steel with an outer diameter of about 3.0 cm and a thickness of about 0.6 cm. Electrical insulator 166 may be magnesium oxide powder with a thickness of about 0.3 cm. Conductive section 170 may couple inner conductor 164 with ferromagnetic section 160 and/or outer conductor 168.

FIG. 12 depicts an embodiment of a temperature limited heater with a ferromagnetic outer conductor. The heater may be placed in a corrosion resistant jacket. A conductive layer may be placed between the outer conductor and the jacket. FIGS. 13 and 14 depict cross-sectional views of the embodiment shown in FIG. 12. Outer conductor 168 may be a 3/4"Schedule 80 446 stainless steel pipe. In an embodiment, conductive layer 172 is placed between outer conductor 168 and jacket 174. Conductive layer 172 may be a copper layer. Outer conductor 168 may be clad with conductive layer 172. In certain embodiments, conductive layer 172 may include one or more segments (e. g. , conductive layer 172 may include one or more copper tube segments). Jacket 174 may be a 1-'/4"Schedule 80 347H stainless steel pipe or a 1-'/2"Schedule 160 347H stainless steel pipe. In an embodiment, inner conductor 164 is 4/0

MGT-1000 furnace cable with stranded nickel-coated copper wire with layers of mica tape and glass fiber insulation. 4/0 MGT-1000 furnace cable is UL type 5107 (available from Allied Wire and Cable (Phoenixville, Pennsylvania) ). Conductive section 170 may couple inner conductor 164 and jacket 174. In an embodiment, conductive section 170 may be copper.

FIG. 15 depicts an embodiment of a temperature limited heater with an outer conductor. The outer conductor may include a ferromagnetic section and a non-ferromagnetic section. The heater may be placed in a corrosion resistant jacket. A conductive layer may be placed between the outer conductor and the jacket. FIGS. 16 and 17 depict cross-sectional views of the embodiment shown in FIG. 15. Ferromagnetic section 160 may be 409,410, or 446 stainless steel with a thickness of about 0.9 cm. Non-ferromagnetic section 162 may be copper with a thickness of about 0.9 cm. Ferromagnetic section 160 and non- ferromagnetic section 162 may be placed in jacket 174. Jacket 174 may be 304 stainless steel with a thickness of about 0.1 cm. Conductive layer 172 may be a copper layer. Electrical insulator 166 may be magnesium oxide with a thickness of about 0.1 to 0.3 cm. Inner conductor 164 may be copper with a diameter of about 1.0 cm.

In an embodiment, ferromagnetic section 160 may be 446 stainless steel with a thickness of about 0.9 cm. Jacket 174 may be 410 stainless steel with a thickness of about 0.6 cm. 410 stainless steel has a higher Curie temperature than 446 stainless steel. Such a temperature limited heater may"contain"current such that the current does not easily flow from the heater to the surrounding formation (i. e. , the Earth) and/or to any surrounding water (e. g. , brine in the formation). In this embodiment, current flows through ferromagnetic section 160 until the Curie temperature of the ferromagnetic section is reached ^-After the Curie temperature of ferromagnetic section 1 : 60 is reached, current flows through conductive layer 172.

The ferromagnetic properties of jacket 174 (410 stainless steel) inhibit the current from flowing outside the jacket and"contain"the current. Jacket 174 may also have a thickness that. provides strength-to the temperature limited heater.

FIG. 18 depicts an embodiment of a temperature limited heater. The heating section of the temperature limited heater may include non-ferromagnetic inner conductors and a ferromagnetic outer conductor. The overburden section of the temperature limited heater may include a non-ferromagnetic outer conductor. FIGS. 19,20, and 21 depict cross-sectional views of the embodiment shown in FIG. 18.

Inner conductor 164 may be copper with a diameter of about 1.0 cm. Electrical insulator 166 may be placed between inner conductor 164 and conductive layer 172. Electrical insulator 166 may be magnesium oxide with a thickness of about 0.1 cm to about 0.3 cm. Conductive layer 172 may be copper with a thickness of about 0.1 cm. Insulation layer 176 may be in the annulus outside of conductive layer 172. The thickness of the annulus may be about 0.3 cm. Insulation layer 176 may be quartz sand.

Heating section 178 may provide heat to one or more hydrocarbon layers in the formation.

Heating section 178 may include ferromagnetic material such as 409 or 410 stainless steel. Heating section 178 may have a thickness of about 0.9 cm. Endcap 180 may be coupled to an end of heating section 178.

Endcap 180 may electrically couple heating section 178 to inner conductor 164 and/or conductive layer 172. Endcap 180 may be 304 stainless steel. Heating section 178 may be coupled to overburden section 182. Overburden section 182 may include carbon steel and/or other suitable support materials. Overburden

section 182 may have a thickness of about 0.6 cm. Overburden section 182 may be lined with conductive layer 184. Conductive layer 184 may be copper with a thickness of about 0.3 cm.

FIG. 22 depicts an embodiment of a temperature limited heater with an overburden section and a heating section. FIGS. 23 and 24 depict cross-sectional views of the embodiment shown in FIG. 22. The overburden section may include portion 164A of inner conductor 164. Portion 164A may be copper with a diameter of about 1.3 cm. The heating section may include portion 164B of inner conductor 164. Portion 164B may be copper with a diameter of about 0.5 cm. Portion 164B may be placed in ferromagnetic conductor 186. Ferromagnetic conductor 186 may be 446 stainless steel with a thickness of about 0.4 cm.

Electrical insulator 166 may be magnesium oxide with a thickness of about 0.2 cm. Outer conductor 168 may be copper with a thickness of about 0.1 cm. Outer conductor 168 may be placed in jacket 174. Jacket 174 may be 316H or 347H stainless steel with a thickness of about 0.2 cm.

In some embodiments, a conductor (e. g. , an inner conductor, an outer conductor, a ferromagnetic conductor) may include two or more different materials. In certain embodiments, a composite conductor may include two or more ferromagnetic materials. In some embodiments, a composite ferromagnetic conductor includes two or more radially disposed materials. In certain embodiments, a composite conductor may include a ferromagnetic conductor and a non-ferromagnetic conductor. In some embodiments, a composite conductor may include a ferromagnetic conductor placed over a non- ferromagnetic core. Two or more materials may be used to obtain a relatively flat electrical resistivity versus temperature. profile in a temperature region below the Curie temperature and/or a sharp decrease in. the electrical resistivity at or near the Curie temperature (e. g., a relatively high turndown ratio).. In'some . cases ; two or more materials may be used to provide. more than one Curie temperature for a temperature limited heater In certain embodiments, a composite electrical conductor may be formed using a billet coextrusion process. A billet coextrusion process may include coupling together two or more electrical conductors at relatively high temperatures (e. g. , at temperatures that are near or above 75% of the melting temperature of a conductor). The electrical conductors may be drawn together at the relatively high temperatures. The drawn together conductors may then be cooled to form a composite electrical conductor made from the two or more electrical conductors. In some embodiments, the composite electrical conductor may be a solid composite electrical conductor. In certain embodiments, the composite electrical conductor may be a tubular composite electrical conductor.

In one embodiment, a copper core may be billet coextruded with a stainless steel conductor (e. g., 446 stainless steel). The copper core and the stainless steel conductor may be heated to a softening temperature in vacuum. At the softening temperature, the stainless steel conductor may be drawn over the copper core to form a tight fit. The stainless steel conductor and copper core may then be cooled to form a composite electrical conductor with the stainless steel surrounding the copper core.

In some embodiments, a long, composite electrical conductor may be formed from several sections of composite electrical conductor. The sections of composite electrical conductor may be formed by a billet coextrusion process. The sections of composite electrical conductor may be coupled together using a welding process. FIGS. 25,26, and 27 depict embodiments of coupled sections of composite electrical conductors. In FIG. 25, core 188 extends beyond the ends of inner conductor 164 in each section of a

composite electrical conductor. In an embodiment, core 188 is copper and inner conductor 164 is 446 stainless steel. Cores 188 from each section of the composite electrical conductor may be coupled together by, for example, brazing the core ends together. Core coupling material 190 may couple the core ends together, as shown in FIG. 25. Core coupling material 190 may be, for example Everdur, a copper-silicon alloy material (e. g. , an alloy with about 3 % by weight silicon in copper).

Inner conductor coupling material 192 may couple inner conductors 164 from each section of the composite electrical conductor. Inner conductor coupling material 192 may be material used for welding sections of inner conductor 164 together. In certain embodiments, inner conductor coupling material 192 may be weld used for welding stainless steel inner conductor sections together. In some embodiments, inner conductor coupling material 192 is 304 stainless steel or 310 stainless steel. A third material (e. g., 309 stainless steel) may be used to couple inner conductor coupling material 192 to ends of inner conductor 164. The third material may be needed or desired to produce a better bond (e. g. , a better weld) between inner conductor 164 and inner conductor coupling material 192. The third material may be non-magnetic to reduce the potential for a hot spot to occur at the coupling.

In certain embodiments, inner conductor coupling material 192 may surround the ends of cores 188 that protrude beyond the ends of inner conductors 164, as shown in FIG. 25. Inner conductor coupling material 192 may include one or more portions coupled together. Inner conductor coupling material 192 may be placed in a clam shell configuration around the ends of cores 188 that protrude beyond the ends of inner conductors 164, as shown in the end view depicted in FIG.. 26., Coupling material 194 may be used to. couple together portions (e. g., halves) of inner conductor coupling. material 192. Coupling material 194 may be the same material as inner conductor coupling material 192 or another material suitable : fbr. coupling : 'together-portions of the inner conductor coupling material.

In some embodiments, a composite electrical conductor may include inner conductor coupling material 192 with 304 stainless steel or 310 stainless steel and inner conductor 164 with 446 stainless steel or another ferromagnetic material. In such an embodiment, inner conductor coupling material 192 may produce significantly less heat than inner conductor 164. The portions of the composite electrical conductor that include the inner conductor coupling material (e. g. , the welded portions or"joints"of the composite electrical conductor) may remain at lower temperatures than adjacent material during application of applied electrical current to the composite electrical conductor. The reliability and durability of the composite electrical conductor may be increased by keeping the joints of the composite electrical conductor at lower temperatures.

FIG. 27 depicts another embodiment for coupling together sections of a composite electrical conductor. Ends of cores 188 and ends of inner conductors 164 are beveled to facilitate coupling together the sections of the composite electrical conductor. Core coupling material 190 may couple (e. g. , braze) together the ends of each core 188. The ends of each inner conductor 164 may be coupled (e. g. , welded) together with inner conductor coupling material 192. Inner conductor coupling material 192 may be 309 stainless steel or another suitable welding material. In some embodiments, inner conductor coupling material 192 is 309 stainless steel. 309 stainless steel may reliably weld to both an inner conductor having 446 stainless steel and a core having copper. Using beveled ends when coupling together sections of a composite electrical conductor may produce a reliable and durable coupling between the sections of

composite electrical conductor. FIG. 27 depicts a weld formed between ends of sections that have beveled surfaces.

A composite electrical conductor may be used as a conductor in any electrical heater embodiment described herein. In an embodiment, a composite electrical conductor may be used as a conductor in a conductor-in-conduit heater. For example, a composite electrical conductor may be used as conductor 146 in FIG. 4. In certain embodiments, a composite electrical conductor may be used as a conductor in an insulated conductor heater. FIG. 28 depicts an embodiment of an insulated conductor heater. Insulated conductor 196 may include core 188 and inner conductor 164. Core 188 and inner conductor 164 may be a composite electrical conductor. Core 188 and inner conductor 164 may be located within insulator 166.

Core 188, inner conductor 164, and insulator 166 may be located inside outer conductor 168. Insulator 166 may be magnesium oxide or another suitable electrical insulator. Outer conductor 168 may be copper, steel, or any other electrical conductor.

In some embodiments, insulator 166 may be an insulator with a preformed shape. A composite electrical conductor having core 188 and inner conductor 164 may be placed inside the preformed insulator.

Outer conductor 168 may be placed over insulator 166 by coupling (e. g. , by welding or brazing) one or more longitudinal strips of electrical conductor together to form the outer conductor. The longitudinal strips may be placed over insulator 166 in a"cigar wrap"method to couple the strips in a widthwise or radial direction (i. e. , placing individual strips around the circumference of the insulator and coupling the individual strips to surround the insulator). The lengthwise ends. of the cigar wrapped strips may. be coupled , to lengthwise ends of other cigarwrapped strips to couple the strips lengthwise along the insulated- conductor.

In some embodiments, jacket l74 may be located outside outer conductor 168, as shown in FIG.

29. In some embodiments, jacket. 174. may be stainless steel (e. g., 304 stainless steel) and outer conductor 168 may be copper. Jacket 174 may provide corrosion resistance for the insulated conductor heater. In some embodiments, jacket 174 and outer conductor 168 may be preformed strips that are drawn over insulator 166 to form insulated conductor 196.

In certain embodiments, insulated conductor 196 may be located in a conduit that provides protection (e. g. , corrosion and degradation protection) for the insulated conductor. FIG. 30 depicts an embodiment of an insulated conductor located inside a conduit. In FIG. 30, insulated conductor 196 is located inside conduit 138 with gap 198 separating the insulated conductor from the conduit.

In some embodiments, a composite electrical conductor may be used to achieve lower temperature heating (e. g. , for heating fluids in a production well or reducing the viscosity of fluids in a wellbore).

Varying the materials of the composite electrical conductor may be used to allow for lower temperature heating. In some embodiments, inner conductor 164 (as shown in FIGS. 25-30) may be made of materials with a lower Curie temperature than that of 446 stainless steel. For example, inner conductor 164 may be an alloy of iron and nickel. The alloy may have between about 30% by weight and about 42% by weight nickel with the rest being iron (e. g. , a nickel/iron alloy such as Invar 36, which is about 36% by weight nickel in iron and has a Curie temperature of about 277 °C). In some embodiments, an alloy may be a three component alloy with, for example, chromium, nickel, and iron (e. g. , an alloy with about 6% by weight chromium, 42% by weight nickel, and 52% by weight iron). An inner conductor made of these types of

alloys may provide a heat output between about 250 watts per meter and about 350 watts per meter (e. g., about 300 watts per meter). A 2.5 cm diameter rod of Invar 36 has a turndown ratio of about 2 to 1 at the Curie temperature. Placing the Invar 36 alloy over a copper core may allow for a smaller rod diameter (e. g., less than 2.5 cm). A copper core may result in a high turndown ratio (e. g. , greater than about 2 to 1).

Insulator 166 may be made of a high performance polymer insulator (e. g. , PFA, PEER) when used with alloys with a low heat output (e. g. , Invar 36).

FIG. 31 depicts an embodiment of a temperature limited heater with a low temperature ferromagnetic outer conductor. Outer conductor 168 may be glass sealing alloy 42-6 (about 42.5 % by weight nickel, about 5.75 % by weight chromium, and the remainder iron). Alloy 42-6 has a relatively low Curie temperature of about 295 °C. Alloy 42-6 may be obtained from Carpenter Metals (Reading, Pennsylvania) or Anomet Products, Inc. In some embodiments, outer conductor 168 may include other compositions and/or materials to get various Curie temperatures. In an embodiment, conductive layer 172 is coupled (e. g. , cladded, welded, or brazed) to outer conductor 168. Conductive layer 172 may be a copper layer. Conductive layer 172 may improve a turndown ratio of outer conductor 168. Jacket 174 may be a ferromagnetic metal such as carbon steel. Jacket 174 may protect outer conductor 168 from a corrosive environment. Inner conductor 164 may have electrical insulator 166. Inner conductor 164 may be stranded nickel-clad copper wire. Electrical insulator 166 may be a mica tape winding with overlaid fiberglass braid.

In an embodiment, inner conductor 164 and electrical insulator 166 are a 4/0 MGT-1000 furnace cable or 3/0 MGT-1000 furnace cable. 4/0 MGT-1000. furnace cable or 3/0 MGT-1000 furnace : cable is available from Allied Wire and Cable (Phoenixville, Pennsylvania). In some. embodiments, a protective braid (e. g.,. stainless steel braid) may be placed over electrical insulator 166. in Conductive section 170 may couple'inner conductor 164 to outer conductor. 168 and/or jacket 174.

In some embodiments, jacket 174 may touch or. electrically contact conductive layer 172 (e. g., if the heater is placed in a horizontal configuration). If jacket 174 is a ferromagnetic metal such as carbon steel with the Curie temperature of the jacket above the Curie temperature of outer conductor 168, current will propagate only on the inside of the jacket so that the outside of the jacket remains electrically safe during operation.

In some embodiments, jacket 174 may be drawn down (e. g. , swaged down in a die) onto conductive layer 172 so that a tight fit is made between the jacket and the conductive layer. The heater may be spooled as coiled tubing for insertion into a subsurface formation wellbore.

In certain embodiments, a copper core may be clad or protected with a relatively diffusion-resistant layer (e. g. , nickel). In some embodiments, a composite inner conductor may include iron clad over nickel clad over a copper core. The relatively diffusion-resistant layer may inhibit migration of copper into other layers of the heater including, for example, an insulation layer. In certain types of heaters, inhibiting migration of copper may inhibit potential arcing during use of the heater. In some embodiments, the relatively impermeable layer may inhibit deposition of copper in a wellbore.

In one heater embodiment, an inner conductor may be a 1.9 cm diameter iron rod, an insulating layer may be 0.25 cm thick magnesium oxide, and an outer conductor may be 0.635 cm thick 347H or 347HH stainless steel. The heater may be energized at line frequency (e. g. , 60 Hz) from a substantially constant current source. Stainless steel may be chosen for corrosion resistance in the gaseous subsurface environment and/or for superior creep resistance at elevated temperatures. Below the Curie temperature,

heat may be produced primarily in the iron inner conductor. With a heat injection rate of about 820 watts/meter, the temperature differential across the insulating layer may be approximately 40 °C. Thus, the temperature of the outer conductor may be about 40 °C cooler than the temperature of the inner ferromagnetic conductor.

In another heater embodiment, an inner conductor may be a 1.9 cm diameter rod of copper or copper alloy such as LOHM (about 94% copper and 6% nickel by weight), an insulating layer may be transparent quartz sand, and an outer conductor may be 0. 635 cm thick 1% carbon steel clad with 0.25 cm thick 310 stainless steel. The carbon steel in the outer conductor may be clad with copper between the carbon steel and the stainless steel jacket. The copper cladding may reduce a thickness of carbon steel needed to achieve substantial resistance changes near the Curie temperature. Heat may be produced primarily in the ferromagnetic outer conductor, resulting in a small temperature differential across the insulating layer. When heat is produced primarily in the outer conductor, a lower thermal conductivity material may be chosen for the insulation. Copper or copper alloy may be chosen for the inner conductor to reduce the heat output from the inner conductor. The inner conductor may also be made of other metals that exhibit low electrical resistivity and relative magnetic permeabilities near 1 (i. e. , substantially non- ferromagnetic materials such as aluminum and aluminum alloys, phosphor bronze, beryllium copper, and/or brass).

In some embodiments, a temperature limited heater may be a conductor-in-conduit heater.

Ceramic insulators or centralizers may be positioned on the inner conductor. The inner conductor may - make sliding electrical. contact with, the outer conduit in a sliding connector section. The sliding-connector section may be located at or near the bottom of the heater.

In certain embodiments, : centralizers may be made of silicon nitride (Si3N4). In some ', embodiments, ^silicon, nitride. may be gas pressure sintered reaction bonded silicon nitride. Gas pressure. sintered reaction bonded silicon nitride is made by sintering the silicon nitride at about 1800 °C in a 1,500 psi (10.3 MPa) nitrogen atmosphere to inhibit degradation of the silicon nitride during sintering. One example of a gas pressure sintered reaction bonded silicon nitride may be obtained from Ceradyne, Inc.

(Costa Mesa, California) as Ceralloy 147-31N. Gas pressure sintered reaction bonded silicon nitride may be ground to a fine finish. The fine finish may allow the silicon nitride to slide easily along metal surfaces without picking up metal particles because of the very low surface porosity of the silicon nitride. Gas pressure sintered reaction bonded silicon nitride is a very dense material with high tensile and flexural mechanical strength. Gas pressure sintered reaction bonded silicon nitride may have high thermal impact stress characteristics. Gas pressure sintered reaction bonded silicon nitride is an excellent high temperature electrical insulator and has about the same leakage current at about 900 °C as alumina (AI203) has at about 760 °C. Gas pressure sintered reaction bonded silicon nitride has a thermal conductivity of about 25 watts per meter-°K, which allows good conduction of heat away from the center conductor of conductor-in- conduit heater applications using centralizers or sliding connectors. Silicon nitride is also a good heat radiator because silicon nitride is optically black (i. e. , promotes efficient black body radiant heat transfer).

Other types of silicon nitride such as, but not limited to, reaction-bonded silicon nitride or hot isostatically pressed silicon nitride may be used. With hot isostatic pressing, granular silicon nitride and additives are sintered at 15,000-30, 000 psi (100-200 MPa) in nitrogen gas. Some silicon nitrides may be

made by sintering silicon nitride with yttrium oxide or cerium oxide to lower the sintering temperature so that the silicon nitride does not degrade (e. g. , release nitrogen) during sintering. Adding too much other material to the silicon nitride may increase the leakage current of the silicon nitride at elevated temperatures compared to purer forms of silicon nitride.

Using silicon nitride centralizers may allow for smaller diameter and higher temperature heaters.

Less of a gap may be needed between a conductor and a conduit because of the excellent electrical characteristics of the silicon nitride (e. g., low leakage current at high temperatures). Silicon nitride centralizers may allow higher operating voltages (e. g. , up to at least about 2500 V) to be used heaters due to the electrical characteristics of the silicon nitride. Operating at higher voltages allows longer length heaters to be utilized (e. g. , at lengths up to at least about 1500 m at about 2500 V).

FIG : 32 depicts an embodiment of a conductor-in-conduit temperature limited heater. Conductor 146 may be coupled (e. g. , cladded, coextruded, press fit, drawn inside) to ferromagnetic conductor 186. In some embodiments, ferromagnetic conductor 186 may be billet coextruded over conductor 146.

Ferromagnetic conductor 186 may be coupled to the outside of conductor 146 so that alternating current propagates only through the skin depth of the ferromagnetic conductor at room temperature. Ferromagnetic conductor 186 may provide mechanical support for conductor 146 at elevated temperatures. Conductor 146 may provide mechanical support for ferromagnetic conductor 186 at elevated temperatures. Ferromagnetic conductor 186 may be iron, an iron alloy (e. g. , iron with about 10% to about 27% by weight chromium for corrosion resistance and lower Curie temperature (e. g. , 446 stainless steel) ), or any other ferromagnetic material. In an embodiment, conductor 146 is copper and ferromagnetic conductor 186 is. 446 stainless steel. Conductor 146 and ferromagnetic conductor 186 may be electrically coupled to conduit 138 with sliding, connector 154. Conduit 138 may be anon-ferromagneticmaterial such as, but not limited to,. 347H stainless steel. In one embodiment, conduit 138 is a 1-t/2"Schedule 80 347H stainless steel pipe. One or more centralizers 202 may maintain the gap between conduit 138 and ferromagnetic conductor 186. In an embodiment, centralizer 202 is made of gas pressure sintered reaction bonded silicon nitride.

FIG. 33 depicts another embodiment of a conductor-in-conduit temperature limited heater.

Conduit 138 may be coupled to ferromagnetic conductor 186 (e. g. , cladded, press fit, or drawn inside of the ferromagnetic conductor). Ferromagnetic conductor 186 may be coupled to the inside of conduit 138 to allow alternating current to propagate through the skin depth of the ferromagnetic conductor at room temperature. Conduit 138 may provide mechanical support for ferromagnetic conductor 186 at elevated temperatures. Conduit 138 and ferromagnetic conductor 186 may be electrically coupled to conductor 146 with sliding connector 154.

FIG. 34 depicts an embodiment of an insulated conductor-in-conduit temperature limited heater.

Insulated conductor 196 may include core 188, electrical insulator 166, and jacket 174. Insulated conductor 196 may be coupled to ferromagnetic conductor 186 with connector 200. Connector 200 may be made of non-corrosive, electrically conducting materials such as nickel or stainless steel. Connector 200 may be coupled to insulated conductor 200 and/or ferromagnetic conductor 186 using suitable methods for electrically coupling (e. g. , welding, soldering, braising). Insulated conductor 196 may be placed along a wall of ferromagnetic conductor 186. Insulated conductor 196 may provide mechanical support for

ferromagnetic conductor 186 at elevated temperatures. In some embodiments, other structures (e. g. , a conduit) may be used to provide mechanical support for ferromagnetic conductor 186.

FIGS. 35 and 36 depict cross-sectional views of an embodiment of a temperature limited heater that includes an insulated conductor. FIG. 35 depicts a cross-sectional view of an embodiment of an overburden section of the temperature limited heater. The overburden section may include insulated conductor 196 placed in conduit 138. Conduit 138 may be 1-'/4"Schedule 80 carbon steel pipe internally clad with copper in the overburden section. Insulated conductor 196 may be a mineral insulated cable.

Conductive layer 172 may be placed in the annulus between insulated conductor 196 and conduit 138.

Conductive layer 172 may be approximately 2.5 cm diameter copper tubing. The overburden section may be coupled to the heating section of the heater. FIG. 36 depicts a cross-sectional view of an embodiment of a heating section of the temperature limited heater. Insulated conductor 196 in the heating section may be a continuation of the insulated conductor from the overburden section. Ferromagnetic conductor 186 may be coupled to conductive layer 172. In certain embodiments, conductive layer 172 in the heating section may be copper drawn over ferromagnetic conductor 186 and coupled to conductive layer 172 in overburden section. Conduit 138 may include a heating section and an overburden section. These two sections may be coupled together to form conduit 138. The heating section may be 1-'/4"Schedule 80 347H stainless steel pipe. An end cap, or other suitable electrical connector, may couple ferromagnetic conductor 186 to insulated conductor 196 at a lower end of the heater (i. e. , the end farthest from the overburden section).

FIGS. 37 and 38 depict cross-sectional views of an embodiment of a) temperature limited heater -that includes an insulated conductor. FIG. 3. 7 depicts a cross-sectional. view of an embodiment of an <BR> <BR> ao oyerburden section of the temperature limited heater. Insulated conductor 196 may include core 188,<BR> <BR> electrical, insulator 166, jacket !'74 : Insulated conductor 196 may have a diameter of about 1. 5 cm.

Core 188 may be copper.. Electrical insulator 166. may be magnesium oxide. Jacket 174 may be copper in the overburden section to reduce heat losses. Conduit 138 may be 1"Schedule 40 carbon steel in the overburden section. Conductive layer 172 may be coupled to conduit 138. Conductive layer 172 may be copper with a thickness of about 0.2 cm to reduce heat losses in the overburden section. Gap 198 may be an annular space between insulated conductor 196 and conduit 138. FIG. 38 depicts a cross-sectional view of an embodiment of a heating section of the temperature limited heater. Insulated conductor 196 in the heating section may be coupled to insulated conductor 196 in the overburden section. Jacket 174 in the heating section may be made of a corrosion resistant material (e. g. , 825 stainless steel). Ferromagnetic conductor 186 may be coupled to conduit 138 in the overburden section. Ferromagnetic conductor 186 may be Schedule 160 409, 410, or 446 stainless steel pipe. Gap 198 may be between ferromagnetic conductor 186 and insulated conductor 196. An end cap, or other suitable electrical connector, may couple ferromagnetic conductor 186 to insulated conductor 196 at a distal end of the heater (i. e. , the end farthest from the overburden section).

In certain embodiments, a temperature limited heater may include a flexible cable (e. g. , a furnace cable) as the inner conductor. For example, the inner conductor may be a 27% nickel-clad or stainless steel-clad stranded copper wire with four layers of mica tape surrounded by a layer of ceramic and/or mineral fiber (e. g. , alumina fiber, aluminosilicate fiber, borosilicate fiber, or aluminoborosilicate fiber). A stainless steel-clad stranded copper wire furnace cable may be available from Anomet Products, Inc.

(Shrewsbury, MA). The inner conductor may be rated for applications at temperatures of up to about 1000 °C. The inner conductor may be pulled inside a conduit. The conduit may be a ferromagnetic conduit (e. g., a 3/4"Schedule 80 446 stainless steel pipe). The conduit may be covered with a layer of copper, or other electrical conductor, with a thickness of about 0.3 cm or any other suitable thickness. The assembly may be placed inside a support conduit (e. g. , a 1-'/4"Schedule 80 347H or 347HH stainless steel tubular). The support conduit may provide additional creep-rupture strength and protection for the copper and the inner conductor. For uses at temperatures greater than about 1000 °C, the inner copper conductor may be plated with a more corrosion resistant alloy (e. g., Incoloy 825) to inhibit oxidation. In some embodiments, the top of the temperature limited heater may be sealed to inhibit air from contacting the inner conductor.

In some embodiments, a ferromagnetic conductor of a temperature limited heater may include a copper core (e. g. , a 1.27 cm diameter copper core) placed inside a first steel conduit (e. g., a 1/2"Schedule 80 347H or 347HH stainless steel pipe). A second steel conduit (e. g. , a 1"Schedule 80 446 stainless steel pipe) may be drawn down over the first steel conduit assembly. The first steel conduit may provide strength and creep resistance while the copper core may provide a high turndown ratio.

In some embodiments, a ferromagnetic conductor of a temperature limited heater (e. g. , a center or inner conductor of a conductor-in-conduit temperature limited heater) may include a heavy walled conduit (e. g. , an extra heavy wall 410 stainless steel pipe). The heavy walled conduit may have a diameter of about 2.5 cm. The heavy walled conduit may be drawn down over a copper rod. The copper rod may have a diameter of about zu cm.. The resulting heater may include a thick ferromagnetic sheath (i. e. , the heavy walled conduit with, for example, about a 2. 6 cm outside diameter after drawing) containing, the copper rod.

The heater may have a turndown ratio of about 8: 1. The thickness of the heavy walled conduit may be selected to inhibit deformation of the heater. A thick ferromagnetic conduit may : provide deformation resistance while adding minimal expense to. the cost of the heater..

In another embodiment, a temperature limited heater may include a substantially U-shaped heater with a ferromagnetic cladding over a non-ferromagnetic core (in this context, the"U"may have a curved or, alternatively, orthogonal shape). A U-shaped, or hairpin, heater may have insulating support mechanisms (e. g. , polymer or ceramic spacers) that inhibit the two legs of the hairpin from electrically shorting to each other. In some embodiments, a hairpin heater may be installed in a casing (e. g. , an environmental protection casing). The insulators may inhibit electrical shorting to the casing and may facilitate installation of the heater in the casing. The cross section of the hairpin heater may be, but is not limited to, circular, elliptical, square, or rectangular.

In some embodiments, a temperature limited heater may include a sandwich construction with both current supply and current return paths separated by an insulator. The sandwich heater may include two outer layers of conductor, two inner layers of ferromagnetic material, and a layer of insulator between the ferromagnetic layers. The cross-sectional dimensions of the heater may be optimized for mechanical flexibility and spoolability. The sandwich heater may be formed as a bimetallic strip that is bent back upon itself. The sandwich heater may be inserted in a casing, such as an environmental protection casing, and may be separated from the casing with an electrical insulator.

A heater may include a section that passes through an overburden. In some embodiments, the portion of the heater in the overburden may not need to supply as much heat as a portion of the heater

adjacent to hydrocarbon layers that are to be subjected to in situ conversion. In certain embodiments, a substantially non-heating section of a heater may have limited or no heat output. A substantially non- heating section of a heater may be located adjacent to layers of the formation (e. g. , rock layers, non- hydrocarbon layers, or lean layers) that remain advantageously unheated. A substantially non-heating section of a heater may include a copper conductor instead of a ferromagnetic conductor. In some embodiments, a substantially non-heating section of a heater may include a copper or copper alloy inner conductor. A substantially non-heating section may also include a copper outer conductor clad with a corrosion resistant alloy. In some embodiments, an overburden section may include a relatively thick ferromagnetic portion to inhibit crushing of the heater in the overburden section.

In certain embodiments, a temperature limited heater may provide some heat to the overburden.

Heat supplied to the overburden may inhibit formation fluids (e. g. , water, gasoline) from refluxing or condensing in the wellbore. Refluxing fluids may use a large portion of heat energy supplied to a target section of the wellbore, thus limiting heat transfer from the wellbore to the target section.

A temperature limited heater may be constructed in sections that are coupled (e. g. , welded) together. The sections may be about 10 m long. Construction materials for each section may be chosen to provide a selected heat output for different parts of the formation. For example, an oil shale formation may contain layers with highly variable richness. Providing selected amounts of heat to individual layers, or multiple layers with similar richness, may improve heating efficiency of the formation and/or inhibit collapse of the. wellbore. A splice section may be formed between the sections, for example, by welding the., inner conductors, filling the splice section with. an'insulator, and then welding the outer conductor : : 'Alternatively ; the heater may be formed from larger diameter tubulars and drawn down to a desired length and diameter. A magnesium oxide insulation layer may, be added by a weld-fill-draw method. (starting from metal strip) or a fill-draw method (starting from tubulars) well known in the industry in the manufacture of mineral insulated heater cables. The assembly and filling can be done in a vertical or a horizontal orientation. The final heater assembly may be spooled onto a large diameter spool (e. g. , about 6 m in diameter) and transported to a site of a formation for subsurface deployment. Alternatively, the heater may be assembled on site in sections as the heater is lowered vertically into a wellbore.

A temperature limited heater may be a single-phase heater or a three-phase heater. In a three- phase heater embodiment, a heater may have a delta or a wye configuration. Each of the three ferromagnetic conductors in a three-phase heater may be inside a separate sheath. A connection between conductors may be made at the bottom of the heater inside a splice section. The three conductors may remain insulated from the sheath inside the splice section.

In some embodiments, a temperature limited heater may include a single ferromagnetic conductor with current returning through the formation. The heating element may be a ferromagnetic tubular (e. g., 446 stainless steel (with 25% chromium and a Curie temperature above about 620 °C) clad over 304H, 316H, or 347HH stainless steel) that extends through the heated target section and makes electrical contact to the formation in an electrical contacting section. The electrical contacting section may be located below a heated target section (e. g. , in an underburden of the formation). In an embodiment, the electrical contacting section may be a section about 60 m deep with a larger diameter wellbore. The tubular in the electrical contacting section may be a high electrical conductivity metal. The annulus in the electrical

contacting section may be filled with a contact material/solution such as brine or other materials that enhance electrical contact with the formation (e. g. , metal beads, hematite). The electrical contacting section may be located in a brine saturated zone to maintain electrical contact through the brine. In the electrical contacting section, the tubular diameter may also be increased to allow maximum current flow into the formation with lower heat dissipation in the fluid. Current may flow through the ferromagnetic tubular in the heated section and heat the tubular.

FIG. 39 depicts an embodiment of a temperature limited heater with current return through the formation. Heating element 212 may be placed in opening 118 in hydrocarbon layer 120. Heating element 212 may be a 446 stainless steel clad over a 304H stainless steel tubular that extends through hydrocarbon layer 120. Heating element 212 may be coupled to contacting element 214. Contacting element 214 may have a higher electrical conductivity than heating element 212. Contacting element 214 may be placed in electrical contacting section 216, located below hydrocarbon layer 120. Contacting element 214 may make electrical contact with the earth in electrical contacting section 216. Contacting element 214 may be placed in contacting wellbore 218. Contacting element 214 may have a diameter between about 10 cm and about 20 cm (e. g. , about 15 cm). The diameter of contacting element 214 may be sized to increase contact area between contacting element 214 and contact solution 220. The contact area may be increased by increasing the diameter of contacting element 214. Increasing the diameter of contacting element 214 may increase the contact area without adding excessive cost to installation and use of the contacting element, contacting wellbore 218, and/or contact solution 220. Increasing the diameter of contacting element 214 may allow í.. sufficient electrical contact to be maintained between the contacting element and electrical contacting . section 216. Increasing the contact area. may'also inhibit evaporation or boiling off of contact, solution 220.

Contacting wellbore 218 may be, for example,, a section about 60 m deep with a larger diameter wellbore than opening 118. The annulus of contacting wellbore 218 may be filled with contact solution.

220. Contact solution 220 may be brine or other material that enhances electrical contact with electrical contacting section 216. In some embodiments, electrical contacting section 216 is a water-saturated zone that maintains electrical contact through the brine. Contacting wellbore 218 may be under-reamed to a larger diameter (e. g. , a diameter between about 25 cm and about 50 cm) to allow maximum current flow into electrical contacting section 216 with low heat output. Current may flow through heating element 212, boiling moisture from the wellbore, and heating until the heat output reduces near or at the Curie temperature.

In an embodiment, three-phase temperature limited heaters may be made with current connection through the formation. Each heater may include a single Curie temperature heating element with an electrical contacting section in a brine saturated zone below a heated target section. In an embodiment, three such heaters may be connected electrically at the surface in a three-phase wye configuration. The heaters may be deployed in a triangular pattern from the surface. In certain embodiments, the current returns through the earth to a neutral point between the three heaters. The three-phase Curie heaters may be replicated in a pattern that covers the entire formation.

FIG. 40 depicts an embodiment of a three-phase temperature limited heater with current connection through the formation. Legs 222,224, 226 may be placed in a formation. Each leg 222,224, 226 may have heating element 212 placed in each opening 118 in hydrocarbon layer 120. Each leg may

have contacting element 214 placed in contact solution 220 in contacting wellbore 218. Each contacting element 214 may be electrically coupled to electrical contacting section 216 through contact solution 220.

Legs 222,224, 226 may be connected in a wye configuration that results in a neutral point in electrical contacting section 216 between the three legs. FIG. 41 depicts an aerial view of the embodiment of FIG. 40 with neutral point 228 shown positioned centrally among legs 222,224, 226.

A section of heater through a high thermal conductivity zone may be tailored to deliver more heat dissipation in the high thermal conductivity zone. Tailoring of the heater may be achieved by changing cross-sectional areas of the heating elements (e. g. , by changing ratios of copper to iron), and/or using different metals in the heating elements. Thermal conductance of the insulation layer may also be modified in certain sections to control the thermal output to raise or lower the apparent Curie temperature zone.

In an embodiment, a temperature limited heater may include a hollow core or hollow inner conductor. Layers forming the heater may be perforated to allow fluids from the wellbore (e. g. , formation fluids, water) to enter the hollow core. Fluids in the hollow core may be transported (e. g. , pumped) to the surface through the hollow core. In some embodiments, a temperature limited heater with a hollow core or hollow inner conductor may be used as a heater/production well or a production well.

In an embodiment, a temperature limited heater may be used in a horizontal heater/production well. The temperature limited heater may provide selected amounts of heat to the"toe"and the"heel"of the horizontal portion of the well. More heat may be provided to the formation through the toe than through the heel, creating a"hot. portion''at the toe and. a"warm portion"at the heel : FIG : 42 depicts electrical resistance versus temperature at various applied electrical currents for a .. 446 stainless. steel rod with a diameter of about 2.5 cin and. a 410 stainless steel rod with a diameter of abou.

2 5 cm.. Curves 230-236 depict resistance profiles as. a ! function of temperature for. the 446 stainless steel rod at 440 amps AC (curve 230), 450 amps AC (curve 232), 500. amps AC (curve 234), and 10 amps DC (curve 236). Curves 238-244 depict resistance profiles as a function of temperature for the 410 stainless steel rod at 400 amps AC (curve 238), 450 amps AC (curve 240), 500 amps AC (curve 242), 10 amps DC (curve 244). For both rods, the resistance gradually increased with temperature until the Curie temperature was reached. At the Curie temperature, the resistance fell sharply. Above the Curie temperature, the resistance decreased slightly with increasing temperature. Both rods show a trend of decreasing resistance with increasing AC current. Accordingly, the turndown ratio decreased with increasing current. In contrast, the resistance gradually increased with temperature through the Curie temperature with an applied DC current.

FIG. 43 depicts electrical resistance versus temperature at various applied electrical currents for a temperature limited heater. The temperature limited heater includes a 4/0 MGT-1000 furnace cable inside an outer conductor of 3/4"Schedule 80 Sandvik (Sweden) 4C54 (446 stainless steel) with a 0.3 cm thick copper sheath welded onto the outside of the Sandvik 4C54. Curves 246 through 264 show resistance profiles as a function of temperature for AC applied currents ranging from 40 amps to 500 amps (246: 40 amps; 248: 80 amps; 250: 120 amps; 252: 160 amps; 254: 250 amps; 256: 300 amps; 258: 350 amps; 260: 400 amps; 262: 450 amps; 264: 500 amps). At lower currents (below 250 amps), the resistance increased with increasing temperature up to the Curie temperature. At the Curie temperature, the resistance fell sharply. At higher currents (above 250 amps), the resistance decreased slightly with increasing temperature

up to the Curie temperature. At the Curie temperature, the resistance fell sharply. Curve 266 shows resistance for an applied DC electrical current of 10 amps. Curve 266 shows a steady increase in resistance with increasing temperature, and little or no deviation at the Curie temperature.

FIG. 44 depicts power versus temperature at various applied electrical currents for a temperature limited heater. Curves 268-276 depict power versus temperature for AC applied currents of 300 amps to 500 amps (268: 300 amps; 270: 350 amps; 272: 400 amps; 274: 450 amps; 276: 500 amps). Increasing the temperature gradually decreased the power until the Curie temperature is reached. At the Curie temperature, the power decreased rapidly.

FIG. 45 depicts electrical resistance versus temperature at various applied electrical currents for a temperature limited heater. The temperature limited heater includes a copper rod with a diameter of about 1.27 cm inside an outer conductor of 1"Schedule 80 410 stainless steel pipe with a 0.15 cm thick copper Everdur welded sheath over the 410 stainless steel pipe. Curves 278-288 show resistance profiles as a function of temperature for AC applied currents ranging from 300 amps to 550 amps (278: 300 amps; 280: 350 amps; 282: 400 amps; 284: 450 amps ; 286: 500 amps ; 288: 550 amps). For these AC applied currents, the resistance gradually increases with increasing temperature up to the Curie temperature. At the Curie temperature, the resistance falls sharply. In contrast, curve 290 shows resistance for an applied DC electrical current of 10 amps. This resistance shows a steady increase with increasing temperature,, and little or no deviation at the Curie temperature.

FIG. 46 depicts data for values of skin depth versus. : temperature for a solid 2.54 cm 410 stainless steel rod. at various applied AC electrical currents.. The skin depth was. calculated using EQN. 2 : (2). N 6 = Rl-Rl g (1.- (1/RAC/RDC)) where'8 is the skin depth,-Ri is the radius of the cylinderj RAc is'the-A (: resistance, and RDc is the DC resistance. : In FIG. 46, curves 292-310 show skin depth profiles as a. function of temperature for applied AC electrical currents over a range of about 50 amps to 500 amps (292: 50 amps; 294: 100 amps; 296: 150 amps; 298: 200 amps; 300: 250 amps; 302: 300 amps; 304: 350 amps ; 306: 400 amps ; 308: 450 amps ; 310 : 500 amps). For each applied AC electrical current, the skin depth gradually increased with increasing temperature up to the Curie temperature. At the Curie temperature, the skin depth increased sharply.

FIG. 47 depicts temperature versus time for a temperature limited heater. The temperature limited heater was about a 2 m long heater that included a copper rod with a diameter of about 1.25 cm inside a 1" Schedule XXH 410 stainless steel pipe and about a 0.13 cm copper sheath. The heater was placed in an oven for heating. Alternating current was applied to the heater when the heater was in the oven. The current was increased over about two hours and reached a relatively constant value of about 400 amps for the remainder of the time. Temperature of the stainless steel pipe was measured at three points at about 0.5 m intervals along the length of the heater. Curve 316 depicts the temperature of the pipe at a point about 0.5 m inside the oven and closest to the lead-in portion of the heater. Curve 314 depicts the temperature of the pipe at a point about 0.5 m from the end of the pipe and furthest from the lead-in portion of the heater.

Curve 312 depicts the temperature of the pipe near a center point of the heater. The point at the center of the heater was further enclosed in about a 30 cm section of 2. 54" thick Fiberfraxg insulation. The insulation was used to create a low thermal conductivity section on the heater (i. e. , a section where heat

transfer to the surroundings is slowed or inhibited (a"hot spot") ). The low thermal conductivity section could represent, for example, a rich layer in a hydrocarbon containing formation (e. g. , an oil shale formation). The temperature of the heater increased with time as shown by curves 312,314, and 316.

Curves 312,314, and 316 show that the temperature of the heater increased to about the same value for all three points along the length of the heater. The resulting temperatures were substantially independent of the added Fiberfrax6D insulation. Thus, the temperature limited heater did not exceed the selected temperature limit in the presence of a low thermal conductivity section.

FIG. 48 depicts temperature versus log time data for a 410 stainless steel rod and a 304 stainless steel rod. At a constant applied AC electrical current, the temperature of each rod increased with time.

Curve 322 shows data for a thermocouple placed on an outer surface of the 304 stainless steel rod and under a layer of insulation. Curve 324 shows data for a thermocouple placed on an outer surface of the 304 stainless steel rod without a layer of insulation. Curve 318 shows data for a thermocouple placed on an outer surface of the 410 stainless steel rod and under a layer of insulation. Curve 320 shows data for a thermocouple placed on an outer surface of the 410 stainless steel rod without a layer of insulation. A comparison of the curves shows that the temperature of the 304 stainless steel rod (curves 322 and 324) increased more rapidly than the temperature of the 410 stainless steel rod (curves 318 and 320). The temperature of the 304 stainless steel rod (curves 322 and 324) also reached a higher value than the temperature of the 410 stainless steel rod (curves 318 and 320). The temperature difference between the 'non-insulated section of the 410 stainless steel rod (curve 320) and the insulated section of the 410 stainless steel rod (curve 318) was less than the temperature difference between the non-insulated section of the 304.- stainless steel rod (curve 324) and the insulated section of the 304 stainless steel rod (curve 322).. The ; temperature of the 304 stainless steel rod was increasing at the termination of the experiment while the temperature of the 410 stainless steel rod ; had leveled out.

A numerical simulation (using the computer program FLUENT) was used to compare operation of temperature limited heaters with three turndown ratios. The simulation was done for heaters in an oil shale formation (Green River oil shale). Simulation conditions were: - 61 in length conductor-in-conduit Curie heaters (center conductor (about 2.54 cm diameter), conduit outer diameter about 7.3 cm) - downhole heater test field richness profile for an oil shale formation - about 16.5 cm diameter wellbores at about 9.14 m spacing between wellbores on triangular spacing - 200 hours power ramp-up time to 820 watts/m initial heat injection rate - constant current operation after ramp up - Curie temperature of 720. 6 °C for heater - formation will swell and touch the heater canisters for oil shale richness greater than 35 gals/ton (0. 141/kg) FIG. 49 displays temperature of the center conductor of a conductor-in-conduit heater as a function of formation depth for a Curie temperature heater with a turndown ratio of 2: 1. Curves 326-348 depict temperature profiles in the formation at various times ranging from 8 days after the start of heating to 675 days after the start of heating (326: 8 days, 328: 50 days, 330: 91 days, 332: 133 days, 334: 216 days, 336:

300 days, 338: 383 days, 340: 466 days, 342: 550 days, 344: 591 days, 346: 633 days, 348: 675 days). At a turndown ratio of 2: 1, the Curie temperature of 720. 6 °C was exceeded after about 466 days in the richest oil shale layers. FIG. 50 shows the corresponding heater heat flux through the formation for a turndown ratio of 2: 1 along with the oil shale richness profile (curve 384). Curves 350-382 show the heat flux profiles at various times from 8 days after the start of heating to 633 days after the start of heating (350: 8 days; 352: 50 days; 354: 91 days; 356: 133 days; 358: 175 days; 360: 216 days; 362: 258 days; 364: 300 days; 366: 341 days; 368: 383 days; 370: 425 days; 372: 466 days; 374: 508 days; 376: 550 days; 378: 591 days; 380: 633 days; 382: 675 days). At a turndown ratio of 2: 1, the center conductor temperature exceeded the Curie temperature in the richest oil shale layers.

FIG. 51 displays heater temperature as a function of formation depth for a turndown ratio of 3: 1.

Curves 386-408 show temperature profiles through the formation at various times ranging from 12 days after the start of heating to 703 days after the start of heating (386: 12 days; 388: 33 days; 390: 62 days; 392: 102 days; 394: 146 days; 396: 205 days; 398: 271 days; 400: 354 days; 402: 467 days; 404: 605 days; 406: 662 days; 408: 703 days). At a turndown ratio of 3: 1, the Curie temperature was approached after about 703 days. FIG. 52 shows the corresponding heater heat flux through the formation for a turndown ratio of 3: 1 along with the oil shale richness profile (curve 432). Curves 410-430 show the heat flux profiles at various times from 12 days after the start of heating to 605 days after the start of heating (410: 12 days, 412: 32 days, 414: 62 days, 416: 102 days, 418: 146 days, 420: 205 days, 422: 271 days, 424: 354 days, 426: 467 days, 428 : 605 days, 430 : 7. 49 days). The center conductor temperature never exceeded the Curie temperature for. the turndown ratio of 3: 1. The center conductor temperature also showed'a relatively. flat temperature profile for the 3. :. 1 turndown ratio. a' FIG. 53. shows heater temperature as a function of-formation depth for a turndown ratio of 4: 1.

Curves 434-454 show temperature profiles through the formation at various times ranging from 12 days after the start of heating to 467 days after the start of heating (434: 12 days; 436: 33 days; 438: 62 days; 440: 102 days, 442: 147 days; 444: 205 days; 446: 272 days; 448: 354 days; 450: 467 days; 452: 606 days, 454: 678 days). At a turndown ratio of 4: 1, the Curie temperature was not exceeded even after 678 days.

The center conductor temperature never exceeded the Curie temperature for the turndown ratio of 4: 1. The center conductor showed a temperature profile for the 4: 1 turndown ratio that was somewhat flatter than the temperature profile for the 3: 1 turndown ratio. The simulations show that the heater temperature stays at or below the Curie temperature for a longer time at higher turndown ratios. For this oil shale richness profile, a turndown ratio of greater than 3: 1 may be desirable.

Analytical solutions for the AC conductance of ferromagnetic materials may be used to predict the behavior of ferromagnetic material and/or other materials during heating of a formation. The AC conductance of a wire of uniform circular cross section made of ferromagnetic materials may be solved for analytically. For a wire of radius b, the magnetic permeability, electric permittivity, and electrical conductivity of the wire may be denoted by A c, and cs, respectively. The parameter,//, is treated as a constant (i. e. , independent of the magnetic field strength).

Maxwell's Equations are: (3) V'. B=0 ; (4) VxE+aBlat=O ;

(5) ##D=# ; and (6) VxH-aDlat=J.

The constitutive equations for the wire are: (7) D=#E,B=#H,J=#E.

Substituting EQN. 7 into EQNS. 3-6, setting #= 0, and writing: (8) E(r,t)=Es(r)ejon and (9) H(r,t)=Hs r)ejon, the following equations are obtained: (10) VeSs =0 ; (11) #xEs+j##Hs=0 ; (12) ##Es=0 ; and (13) #xHs-j##Es=#Es.

Note that EQN. 12 follows on taking the divergence of EQN. 13. Taking the curl of EQN. 11, using the fact that for any vector function F: and applying EQN. 10, it is deduced that: with (17) Se¢ =#+j## For a cylindrical wire, it is assumed that: (18) Es = Es (r) k which means that Es (r) satisfies the equation: The general solution of EQN. 19 is: (20) Es (r) = AIo (Cr) + BKo (Cr).

B must vanish as Ko is singular at r = 0, and so it is deduced that: The power output in the wire per unit length (P) is given by: and the mean current squared (<12>) is given by: EQNS. 22 and 23 may be used to obtain an expression for the effective resistance per unit length (R) of the wire. This gives:

with the second term on the right-hand side of EQN. 24 holding for constant a.

C may be expressed in terms of its real part (CR) and its imaginary part (CI) so that: (25) C = CR + iCI.

An approximate solution for CR may be obtained. CR may be chosen to be positive. The quantities below may also be needed: (26) 1 C I {CR + Cl} and (27) ##C/#C#=#R+i#I.

A large value of Re (z) gives: This'means that: (29) Es (r)#Es(b)e-##, with (30) 4 =1 C | (b-r) Substituting EQN. 29 into EQN. 24 yields the approximate result: EQN. 31 may be written in the form: (32) R=1/(2#b##) , Eis known as the skin depth, and the approximate form in EQN. 33 arises on replacing #eff by #.

The expression in EQN. 29 may be obtained directly EQN. 19. Transforming to the variable gives: The solution of EQN. 34 can be written as:

The solution of EQN. 37 is: (39) Es(0)=Es(a)e-##, and solutions of EQN. 38 for successive m may also be readily written down. For instance: (40) Es(1)=#Es(a)#e-#.

The AC conductance of a composite wire having ferromagnetic materials may also be solved for analytically. In this case, the region 0 < r < a may be composed of material 1 and the region a < r < b may be composed of material 2. Esl (r) and Es2 (r) may denote the electrical fields in the two regions, respectively. This gives: with (43) Ck =j##k#effk; k= 1,2 and (44)#effk =#k+j##k ; k = 1,2.

The solutions of EQNS. 41 and 42 satisfy the boundary conditions : (45) Es1(a)=ES2 (a) and (46) Hsl (a) = Hs2 (a) and take the form: (47) Es, (r) =A1I0(C1r) and (48) ES2 (r) =A2I0(C2r)+B2K0(C2r).

Using EQN. 11, the boundary condition in EQN. 46 may be expressed in terms of the electric field as: Applying the two boundary conditions in EQNS. 45 and 49 allows Esl (r) and Es2 (r) to be expressed in terms of the electric field at the surface of the wire ES2 (b). EQN. 45 yields: (50) Al =A2I0(C2a)+B2K0(C2a), while EQN. 49 gives: (51) A1C1I1(C1a) = C2 {A2I1(C2a)-B2K1(C2a)}.

Writing EQN. 51 uses the fact that: and introduces the quantities: (53) C, =C ; C, =C,/.

Solving EQN. 50 for A2 and B2 in terms of Al obtains: Power output per unit length and AC resistance of a composite wire may be solved for similarly to the method used for the uniform wire. In some cases, if the skin depth of the conductor is small in comparison to the radius of the wire, the functions containing C2 may become large and may be replaced by exponentials. However, as the temperature nears the Curie temperature, a full solution may be required.

The dependence of µ on B may be treated iteratively by solving the above equations first with a constant/ a to determine B. Then the known B versus H curves for the ferromagnetic material may be used to iterate for the exact value often the equations.

Further modifications and alternative embodiments of various aspects of the invention may be apparent to those skilled in the art in view of this description. Accordingly, this description is'to be construed as illustrative only and. is for the purpose of teaching those skilled in the art the general manner of carrying out the invention. It is to be understood that the forms of the invention shown and described herein are to be taken as the presently preferred embodiments. Elements and materials may be substituted for those illustrated and described herein, parts and processes may be reversed, and certain features of the invention may be utilized independently, all as would be apparent to one skilled in the art after having the benefit of this description of the invention. Changes may be made in the elements described herein without departing from the spirit and scope of the invention as described in the following claims. In addition, it is to be understood that features described herein independently may, in certain embodiments, be combined.