1

A Simple Model of Gravitationally-Driven Water Flow in a Semicircular 2

Aquifer to Estimate Thermal Power Potential: Application to Clifton Hot 3

Springs, Eastern Arizona 4

5

Paul Morgan 6

7

Department of Geology 8

Box 4099, Frier Hall, Knoles Drive 9

Northern Arizona University 10

Flagstaff, AZ 86011-4099 11

USA 12

13

Paul.Morgan@nau.edu 14

15

Submitted for Consideration for Publication in 16

Journal of Volcanology and Geothermal Research 17

18

October 2005 19

A Simple Model of Gravitationally-Driven Water Flow in a Semicircular 1

Aquifer to Estimate Potential Thermal Power Potential: Application to 2

Clifton Hot Springs, Eastern Arizona 3

4

Abstract 5

6

An analytical hot-spring model has been developed that is a simplification of 7

water flow in a vertical fracture system, and the model is applied to data from the Clifton 8

Hot Springs in eastern Arizona. The model assumes that the path of water flow driven by 9

piezometric (gravity-induced) gradients in near-vertical fractures may be approximated 10

by a simple semicircular path with uniform cross-section along the path. The solution is 11

developed to show that with information about the system that can be obtained at the 12

system’s discharge, including water outflow temperature, mean surface temperature, and 13

water geochemistry, the thermal power potential of the system can be calculated. 14

15

The Clifton Hot Springs have an outflow temperature of 30 to 70°C at a mean 16

surface temperature of 19°C and a maximum aquifer temperature of 180°C. From these 17

data an aquifer flow rate of 19 l s-1 (liters per second) has been calculated which 18

compares well with a published flow rate of 76 l s-1 calculated from the chloride budget 19

of the San Francisco River into which the springs discharge. The maximum thermal 20

output of the springs at their outflow temperature is calculated to be within the range of 4 21

to 16 MW, but this is only at a temperature suitable to direct use. If the system can be 22

tapped at its maximum temperature its maximum thermal output rises to 11 to 44 MW 1

and temperatures are high enough for electricity generation. However, not all of this 2

thermal power output is available for electricity generation. A more conservative 3

estimate of economically attainable electrical power from this system would be 1-3 MW. 4

5

Keywords 6

Geothermal; Aquifer; Hot Springs; Groundwater Temperature; Water Flow; 7

Geothermometry 8

Introduction 1

2

Hot springs, geysers, fumaroles, and other surface thermal features are the surface 3

manifestations of subsurface geothermal systems. These subsurface systems may be 4

driven by 1) localized magmatic heating of the upper crust or 2) heating of groundwater 5

in permeable upper crustal rocks by regional heat flow and the resulting regional 6

geothermal gradient which may be locally increased by low thermal conductivity rocks 7

where the circulation occurs. If the geothermal gradient is sufficiently high, and the 8

permeability sufficiently low, groundwater circulation may be driven by thermal 9

convection. More commonly, heat is transferred by groundwater movements driven 10

primarily by piezometric gradients (caused by changes in the elevation of the water table) 11

with thermal buoyancy forces a secondary factor. 12

13

There are two end members for models of geothermal systems. At one extreme, 14

models seek to be as realistic as possible including everything that is known about the 15

geology of the systems, the physics and the chemistry of the heat transfer and fluid 16

behavior in the system. These models are typically complex numerical computer 17

simulations. At the other extreme are analytical simplifications of the plumbing, flow, 18

and heat transfer within the systems, often completely ignoring the chemistry of the 19

systems. These models may use dimensionless parameterization of the systems so that 20

they may be applied to a wide range of scales without repeated computations. They are 21

not expected to be an accurate simulation of any particular geothermal systems, but they 22

provide useful order-of-magnitude estimates of the thermal parameters of classes of 1

geothermal systems. 2

3

In this contribution, an analytical hot-spring model is developed that is a 4

simplification of water flow in a vertical fracture system, and the model is applied to data 5

from the Clifton Hot Springs in eastern Arizona. The model assumes that the path of 6

water driven flow driven by piezometric (gravity-induced) gradients in near-vertical 7

fractures may be approximated by a simple geometrically-shaped path that has a uniform 8

cross-section along the path. The basic solution for the shape chosen, a semi-circle, has 9

already been solved, but requires information about the system, such as maximum depth 10

of water flow, that is not readily available. However, the solution is developed further 11

here to show that with only information about the system that can be collected at the 12

system’s discharge, including water outflow temperature, mean surface temperature, and 13

major element chemistry of the outflow waters, the basic thermal parameters, including 14

the thermal power potential of the system, can be calculated. 15

16

Semicircular Aquifer Model 17

18

Groundwater flow is assumed to be driven by piezometric (gravitationally-19

derived) pressure gradients usually associated with topography and recharge. Recharge 20

to the groundwater system is typically greatest in areas of high elevation resulting in a 21

high water table and high piezometric pressure, and water flows to areas of lower 22

elevations where the water table is lower and piezometric pressures are lower. In a 1

homogeneous earth, the resulting flow would be greatest at the water table, diminishing 2

in magnitude with depth. In a heterogeneous earth, flow is concentrated through aquifers 3

and permeable fracture zones and diverted around aquitards. 4

5

Hot springs are formed where groundwater flows to depth before returning to the 6

surface. Such flow may be part of a large basin system, for example, in the hot springs 7

between the basins of the Rio Grande rift (Morgan et al., 1981), or in more restricted 8

vertical fracture systems, as with the Clifton Hot Springs. In both types of flow systems a 9

significant component of deep flow in the system must be present, and the velocity of this 10

flow must be balanced: too high of a velocity results in cooling of the rocks at depth and 11

too slow of a velocity results in cooling of the groundwater as it returns to the surface. 12

Simple models may be used to estimate the effective ranges of flow velocities to produce 13

hot springs in flow systems with different geometries (e.g., Morgan et al., 1981). 14

15

The simplest geometry that may be used to model flow in a vertical fracture 16

assumes that the permeable zone, or aquifer, is circular in cross section and semi-circular 17

in its path, as shown in Figure 1. These assumptions are clearly simplistic. However, 18

they allow testing of the relative importance of basic parameters such as geothermal 19

gradient, aquifer surface area, and flow rate. 20

21

==== Figure 1 about here ==== 22

1

Assuming a vertical thermal gradient in the earth, β, water is heated as it descends 2

through the aquifer and cools as it ascends to the surface, depending on the water flow 3

rate. This heating and cooling may be demonstrated for the general case by formulating 4

the problem in dimensionless numbers and assuming laminar flow, as shown in Figure 2 5

(see Appendix for the formulae to calculate these temperatures). The maximum 6

temperature in the aquifer and the exit temperature from the aquifer (hot spring 7

temperature) depend on the flow rate, as shown in Figure 3 (formulae for calculation of 8

these temperatures are also given in the Appendix). 9

10

=== Figures 2 and 3 about here === 11

12

Figures 2 and 3 are plotted using dimensionless temperature θ on the ordinate (y) 13

axes and this is defined as: 14

( )( ) ,'

)

sR

s

TTTT−

−=θ (1) 15

16

where T is temperature, Ts is surface temperature, and TR is the undisturbed wall-rock 17

temperature at the maximum depth penetrated by the aquifer (controlled by the 18

geothermal gradient). In Figure 3 the dimensionless temperature on the ordinate axis is 19

the water exit (hot spring) temperature, θe. 20

21

Dimensionless flow rate is defined by (r Pe)/R, where Pe is the Péclet number 1

defined as: 2

3

κruPe = (2) 4

5

where ū is mean laminar flow velocity in the aquifer, R is the radius of curvature of the 6

semicircular aquifer, κ is the thermal diffusivity, and the aquifer has a circular cross 7

section of radius r. 8

9

For a typical hot spring, parameters that are easily measured or available are 10

outflow temperature, TO, and surface temperature (mean annual surface temperature, Ts). 11

Water geochemistry may be used to estimate the reservoir temperature of the aquifer 12

feeding the hot spring, Tm, (Fournier and Rowe, 1966, Fournier and Truesdell, 1973, 13

Fournier, 1977), and as fracture systems are unlikely to have large-volume reservoirs, the 14

“reservoir temperature” estimated by groundwater geochemistry is likely to be close to 15

the maximum temperature reached by the groundwater flowing through the fracture 16

system As shown in Figure 2, this maximum temperature decreases with increasing flow 17

rate, (r Pe)/R, and the position at which the groundwater reaches its maximum 18

temperature moves toward the flow outlet as flow rate increases. We can determine the 19

maximum dimensionless temperature for each dimensionless flow rate (see Appendix) 20

shown in Figure 2 and make a third dimensionless parameter plot of the ratio of 21

dimensionless exit temperature θe (outflow temperature minus mean annual surface 22

temperature defined as in Equation 1) to maximum dimensionless temperature θmax 1

versus dimensionless flow rate. Taking the ratio of the dimensionless exit temperature to 2

the maximum temperature removes the need to know the maximum wall-rock 3

temperature in calculation of the dimensionless temperatures as it cancels when the ratio 4

is taken (Equation 3). This plot is shown in Figure 4. 5

6

Ratio of dimensionless exit temperature to dimensionless maximum temperature = 7

( )( )( )

( )( )sm

sO

sR

sm

sR

sOe

TTTT

TTTT

TTTT

−−

=−−

−−

=)(

maxθθ

(3) 8

9

=== Figure 4 about here === 10

11

How may these dimensionless plots be applied to the real world? 12

13

For most hot springs the mean annual surface temperature is already known, or it 14

can be measured. The outflow temperature can be measured. The maximum reservoir 15

temperature can be determined from water geochemistry. Using these data, the ratio of 16

dimensionless exit temperature to dimensionless maximum temperature (Equation 3) may 17

be used to estimate the dimensionless flow rate from Figure 4. Using this estimate of 18

dimensionless flow rate, the dimensionless maximum temperature in the aquifer θmax and 19

exit temperature θe from the system may be determined from either Figure 3, or 20

polynomial fits to these curves. Thus, using either θmax or θe and the relations give in 21

Equation 3, the maximum undisturbed wall rock temperature penetrated by the aquifer, 1

TR, may be calculated. 2

3

In most thermal areas where the hot springs are not thought to have a volcanic 4

heat source, some estimate of the regional geothermal gradient or heat flow is available. 5

If the geothermal gradient, ∂T/∂z, is available it may be used directly. If a local heat flow 6

value is available it generally has an associated geothermal gradient, or the geothermal 7

gradient may be estimated by dividing the heat flow by the average thermal conductivity 8

of the rocks associated with the flow system of the hot spring. An estimate of the 9

maximum depth of penetration of the aquifer, R, may then be determined by dividing the 10

maximum undisturbed rock temperature penetrated by the aquifer by the geothermal 11

gradient, or: 12

TzTR R ∂

∂= . (4) 13

14

If ψ is the dimensionless flow rate, (R Pe)/R, which has previously been 15

determined, then by substituting the definition of the Péclet number from Equation 2 we 16

obtain: 17

18

Rru

R

ruR

κκψ

2

== , (5) 19

20

The dimensional flow rate is given by the linear (mean) velocity in the aquifer, ū, 1

multiplied by the cross-sectional area of the aquifer, π r2 (the aquifer was defined to have 2

a circular cross section of radius r). If there is no loss of water from the aquifer, the 3

discharge rate from the aquifer is equal to the volume flow rate in the aquifer, which is 4

given by: 5

6

Aquifer discharge rate, V = Rru κπψπ =2 (6) 7

8

The thermal diffusivity of crystalline rocks is typically about 1 mm2 s-1. If the thermal 9

diffusivity of the rocks penetrated by the aquifer is known more accurately from other 10

sources, the more accurate value may be used, but thermal diffusivity is usually in the 11

range of 0.5 to 2.0 mm2 s-1 for most rocks and minerals that are likely to be hosts for 12

fracture flow in the temperature range of 20 to 120°C (Roy et al., 1981) and the use of 1 13

mm2 s-1 will not result in serious errors with respect to the order of magnitude estimates 14

that are the goal of these calculations. Thus, using the equations above and the 15

assumptions of a semicircular aquifer, if the hot spring temperature and mean annual 16

surface temperature can be measured, and water geochemistry can be used to estimate the 17

maximum reservoir temperature, the total aquifer discharge can be estimated. 18

Alternatively, if the total aquifer discharge can also be measured, the assumptions and 19

applicability of the semicircular aquifer model may be tested. 20

21

If the semicircular aquifer model is considered to be a useful approximation to the 1

hot spring system under study on the basis of these calculations, then the following 2

information results from the calculations: 3

4

1. The absolute maximum rate of thermal energy extraction from exiting hot 5

string waters is the product of the difference between the hot spring temperature and the 6

mean annual surface temperature, (TO-Ts), the discharge rate of the hot springs, V , and 7

the specific heat of water, c: 8

9

Maximum thermal extraction rate from hot springs = (TO -Ts) V c. (7) 10

11

In practice, the available thermal energy will be less. 12

13

2. The absolute maximum rate of thermal extraction of the maximum temperature 14

of the aquifer system that can be tapped is the product of the difference between the 15

maximum aquifer temperature and the mean annual surface temperature, (Tm-Ts), the 16

discharge rate of the hot springs, V, and the specific heat of water, c: 17

18

Maximum thermal extraction rate from hot springs = (Tm -Ts) V c. (8) 19

20

Again, in practice, the available thermal energy will be less. 21

22

3. The location of the maximum aquifer temperature is generally significantly 1

displaced from the discharge point of the hot springs both horizontally and in depth (see 2

equations above and Figures 2, 3, and 4). A fracture aquifer may be a difficult 3

exploration and drilling target especially if the aquifer is not semicircular, but more 4

elongate in shape, and if the fractures controlling the flow do not have continuous surface 5

expression from the recharge area to the hot springs. 6

7

Application to the Clifton Hot Springs, Eastern Arizona 8

9

Clifton Hot Springs are located adjacent to the San Francisco River, just north of 10

the town of Clifton in Safford County in the southeastern quadrant of Arizona (Figures 11

5). The surface manifestations of the Clifton geothermal system are numerous springs 12

and seeps along the San Francisco River with temperatures ranging between 30 and 70ºC 13

(Witcher, 1981). A number of similar hot spring manifestations within a 40 km radius of 14

Clifton are also present, mostly to the northeast. The Clifton Hot Springs are in an area 15

that has previously been defined as a KGRA (Known Geothermal Resource Anomaly), 16

and two of these other local surface manifestations are also undeveloped KGRAs. The 17

hottest springs in Arizona are Gillard Hot Springs, about 6 miles (10 km) south-southeast 18

of Clifton, which discharge 80 to 84ºC water along the banks of the Gila River (Witcher, 19

1981). Eagle Creek Hot Springs to the west of Clifton have a temperature of 42ºC, and a 20

warm spring, Hanna Creek Hot Springs, and Lower Frisco Hot Springs, all to the 21

northeast of Clifton, have temperatures of 26.6ºC, 60ºC, and 49ºC, respectively (Witcher, 1

1981). 2

3

Geologic and Geographic Setting 4

5

The Clifton geothermal system sits on the edge of a large sheet of mid-Tertiary 6

(middle Miocene to Oligocene: 15 to 38 Ma – million years ago) volcanic rocks at the 7

transition into a basin filled with sedimentary and volcanic rocks of a similar age 8

(http://geology.asu.edu/~reynolds/azgeomap/ azgeomap_home.htm, accessed 2005-9-14). 9

The mid-Tertiary volcanic rocks adjacent to the Clifton geothermal system range from 10

silicic to mafic in composition. They are associated with a very strong crustal heating 11

event that swept east, then back west across southern Arizona and New Mexico during 12

the time period of 38 to 15 Ma, and sometimes called the “ignimbrite flare-up” because 13

of the large number of silicic ash flows that were erupted during this event. These ash 14

flows were associated with the emplacement of granitic bodies in the upper crust, many 15

of which are now exposed in southern Arizona. In general, such bodies are the targets of 16

geothermal exploration. However, the “ignimbrite flare-up” event is too old to be 17

associated with modern economic geothermal resources. High-temperature geothermal 18

resources are typically associated with emplacement of upper crustal silicic bodies 19

younger than a few tens of thousands to around a hundred thousand years in age. Thus, 20

the mid-Tertiary volcanic rocks in this region are not considered to be viable heat sources 21

for the hot and warm springs in the Clifton geothermal system, nor in any of the nearby 22

geothermal systems. The “ignimbrite flare-up” and the tectonic (faulting) events that 1

followed have, however, resulted in a regional background crustal heat flow and 2

geothermal gradient that is one and one half to two times the heat flow and geothermal 3

gradient that would typically be measured in a region that had not had any volcanic or 4

tectonic activity for 150 Ma or more (Morgan, 2002). 5

6

Surface topography is important in understanding geothermal systems because 7

rainfall is often elevation dependant and subsurface water flow is controlled by the slope 8

of the water table, which is generally a subdued representation of the topography. 9

Arizona has three basic physiographic provinces: the Colorado Plateau, which covers 10

northeastern Arizona and the adjacent quadrants of New Mexico, Colorado, and Utah; the 11

southern Basin and Range province, a dissected extended terrain surrounding the 12

Colorado Plateau to the south and west in Arizona; and the Transition Zone, a belt of 13

intermediate physiography that separates the Colorado Plateau and Basin and Range 14

provinces in Arizona. 15

16

The surface manifestations of the Clifton geothermal system, the hot/warm 17

springs, emerge in valleys cut into the mid-Tertiary volcanic rocks on the southern edge 18

of the Colorado Plateau. A similar relation is found for all of the surface geothermal 19

manifestations in the region. The margin of the Colorado Plateau is generally elevated 20

relative to its interior and drains inward toward the Colorado River system. The southern 21

margin of the Plateau in Arizona is known as the Mogollon Rim and was once a tectonic 22

(faulted) margin, but is now being eroded back by streams draining to the south. As these 1

streams cut into the Plateau they also capture some of the groundwater from the Plateau, 2

and this groundwater feeds the springs on the southern margin of the Plateau. Springs are 3

also developed near the bases of the ranges on the southern margin of the Transition Zone 4

and near the bases of some of the mountain ranges in the Basin and Range Province to 5

the south. However, in general, groundwater in these systems has neither the elevation 6

drop, nor the distance of horizontal travel that is available to groundwater coming in from 7

the Mogollon Rim. 8

9

The observation of the correlation of the Clifton geothermal system and other 10

nearly geothermal features with the Mogollon Rim suggests that there should be other 11

geothermal systems in Arizona associated with the Rim. However, there is a subtle 12

change in the physiography and geologic structure to the west. The topography becomes 13

less rugged, and the Transition Zone becomes more distinct. The Transition Zone is 14

basically a faulted basin parallel to the Mogollon Rim and acts as a buffer zone between 15

the Basin and Range province and the Colorado Plateau. It represents a mid-Tertiary 16

period of extensional faulting when northwest-southeast basins were formed by 17

northeast-southwest forces. To the east in New Mexico these basins ran directly into the 18

southwest-northeast margin of the Colorado Plateau, and no Transition Zone was formed, 19

but extensive volcanism occurred forming the Datil-Mogollon volcanic field. The mid-20

Tertiary volcanism in the Clifton area, discussed above, is part of the Datil-Mogollon 21

volcanism, and the Colorado Plateau in this region is essentially open to the roughly 22

north-south trending topography of the later extensional faulting of the Basin and Range 1

to the south. This young north-south faulting actually penetrates into the Clifton area 2

(Witcher and Stone, 1980) 3

4

Geothermal features in the Clifton region continue eastward into New Mexico 5

(Lower Frisco Hot Springs KGRA). There are at least fourteen hot springs in New 6

Mexico in the Datil-Mogollon volcanic field with temperatures of 31ºC or above 7

(maximum temperature 87ºC) between longitudes 108ºW and 109ºW, and latitudes 33ºN 8

and 34ºN, east of the Arizona-New Mexico state line (National Geophysical Data Center / 9

WDC-A for Solid Earth Geophysics Boulder, Colorado USA, Hot Springs in the U.S., 10

New Mexico, 11

http://www.ngdc.noaa.gov/nndc/struts/results?op_0=eq&v_0=NM&op_1=l&v_1=&op 12

_2=l&v_2=&t=100006&s=1&d=1, accessed 2005-9-14). Thus, physiographically, 13

geologically, and probably geothermally, the Clifton geothermal system is part of the 14

geothermal systems encompassed by the Datil-Mogollon volcanic field, and has more in 15

common with the large number of geothermal systems to the east in New Mexico, rather 16

than to a general Arizona Mogollon-Rim Transition-Zone-type geothermal system. 17

18

Water Chemistry and Geothermometry 19

20

Water emerging at the surface from hot springs always experiences some degree 21

of cooling from its reservoir temperature at depth. Even if ascent rates are very rapid, 22

cooling occurs due to adiabatic decompression, but generally some temperature loss also 1

occurs due to conduction. A temperature drop may also occur from mixing with cooler 2

waters or from steam release. 3

4

Information concerning reservoir temperatures at depth may be obtained using 5

chemistry of the water emerging at the surface providing a number of conditions are met: 6

a) temperature-dependent reactions control the chemistry of waters at equilibrium in the 7

geothermal reservoirs and an adequate supply of all chemical constituents for the 8

chemical reactions is available; b) no temperature-controlled re-equilibration of the 9

chemical composition of the waters occur after they leave the reservoirs and cool 10

conductively on the way to the surface; c) no mixing with other waters or steam loss 11

occurs before the water discharge at the surface, or if mixing or steam loss occurs, no 12

precipitation of chemical components occurs and some method of estimating the amount 13

of mixing or steam loss exists. 14

15

The two primary methods of estimating reservoir temperatures using water 16

chemistry are the silica and the Na-K-Ca (sodium-potassium-calcium) geothermometers. 17

The water solubility of different varieties of silica, quartz, chalcedony, and opal, are 18

temperature dependent and minimum subsurface temperatures may be calculated using 19

the concentration of silica in spring or flowing well waters (Fournier and Rowe, 1966). 20

Silica does not immediately precipitate when a relatively silica-rich hot water cools (the 21

silica forms quasi-stable silica complexes), and thus a hot spring water typically contains 22

most of the silica concentration dissolved while in its geothermal reservoir (Fournier, 1

1977). This geothermometer is most applicable where the geothermal water is not mixed 2

and diluted with cold, low-silica waters. 3

4

The Na-K-Ca geothermometer is based on the temperature-dependant equilibrium 5

of water with the common mineral feldspar. It is an empirical relation based on the 6

proportions of potassium to sodium, the square root of calcium to sodium, and measured 7

temperature in geothermal wells (Fournier and Truesdell, 1973). The application of this 8

geothermometer is more complicated than the silica geothermometer, and when 9

significant concentrations of magnesium are present, a magnesium correction is 10

necessary. However, this geothermometer is less affected by mixing if the concentrations 11

of sodium, potassium and calcium are high in the geothermal reservoir water relative to 12

the mixing water as the geothermal reservoir water will continue to dominate the 13

proportions of these ions after mixing. 14

15

Waters discharging from springs in the Clifton geothermal system have silica 16

concentrations ranging from 39 to 131 mg/l which correspond to reservoir temperatures 17

of 91 to 153ºC assuming reservoir water equilibrium with quartz. The average of these 18

temperatures is lowered by about 25ºC if the equilibrium is assumed to be with 19

chalcedony (Witcher 1979, 1981). Na-K-Ca geothermometer calculations based on 20

chemistry of waters from the Clifton Hot Springs predict reservoir temperatures between 21

159 and 191ºC, with the majority in the 170 to 180ºC range (Witcher, 1981) These 22

predicted reservoir temperatures are higher than those predicted by the silica 1

geothermometer, but, as explained above, the Na-K-Ca geothermometer is less 2

susceptible to the effects of mixing than the silica geothermometer (the springs have high 3

ion contents, up to 14,500 mg/l total dissolved solids). Witcher (1981) presented strong 4

evidence that the Clifton Hot Spring waters are mixed with at least one component of 5

cold water before emerging at the surface, and he also presented a plausible mixing 6

model that corrects the silica geotemperature data to be compatible with the higher Na-K-7

Ca reservoir temperature data. For more details the reader is referred to Witcher (1981), 8

170 to 180ºC is accepted as the best range for the estimated maximum reservoir 9

temperature for the Clifton geothermal system based on available data. 10

11

Heat Flow and Other Geophysics 12

13

Water chemistry and geothermometry give important information about potential 14

maximum reservoir temperatures, but yield no direct spatial information concerning the 15

locations of these reservoirs. To tap higher temperatures in the geothermal systems, 16

some information and/or understanding of subsurface control of the system must be 17

obtained so that if temperatures and/or fluid flow rates at the hot springs are insufficient 18

for economical use of the thermal resource, wells may be drilled in a part of the system 19

where temperatures/flow rates allow economic recovery of the thermal resource. This 20

additional information and understanding may come from a combination of geologic, 21

hydrologic, and geophysical studies. The geologic information with respect to heat 22

sources has been discussed above together with topographic arguments related to 1

subsurface water flow. In this section geophysical studies of the Clifton geothermal 2

system and surrounding region are discussed, and in particular heat flow studies, and 3

their integration with geology and hydrology in an attempt to understand the subsurface 4

structure of the hydrothermal system(s). 5

6

Gravity, electrical, passive and active seismic studies have been made of the 7

shallow and deep crust in the region of the Clifton geothermal system. Specifically, 8

gravity and electrical (audio-magnetotelluric or AMT) surveys were made by the U. S. 9

Geological Survey as part of their reconnaissance evaluation of the potential economic 10

geothermal potential of the Gillard and Clifton Hot Springs KGRAs. In the complete 11

Bouguer gravity map of the Clifton area, a steep gravity gradient strikes west-northwest 12

across the area, passing just north of Clifton, and separates relatively high gravity in the 13

south from lower gravity in the north. This gradient appears to represent a significant 14

crustal discontinuity (Boler et al., 1980; Witcher et al., 1981). Other areas of high and 15

low gravity on the map are interpreted to be caused by thick and low-density basin-filling 16

sediments, thick piles of volcanics, or intrusions (Klein et al, 1980; Witcher et al., 1981), 17

but do not appear to correlate directly with the modern geothermal systems. Electrical 18

data from close to the Clifton Hot Springs show low apparent resistivity values at shallow 19

depth, corresponding to the hot, relatively saline waters of the springs. On a Precambrian 20

granite bedrock site just north of the springs much higher resistivities were measured, 21

normal for granite, except for a channel of slightly lower resistivity that could represent 22

hydrothermal alteration, groundwater, or a lateral flow of relatively fresh water at a 1

temperature of less than 150°C (Klien et al., 1980, Witcher, 1981). Thus, the electrical 2

data do not trace the hot, relatively saline water of the hot springs outside the immediate 3

area of emergence of the springs. 4

5

Passive and active seismic studies provide large-scale information about crustal 6

structure. A short-term (two week) local study of seismicity in the area surrounding 7

Clifton in 1978 using highly-sensitive portable seismographs showed no local 8

microearthquake activity (P. Morgan, G. R. Keller, and M. Sbar, unpublished data), 9

which, combined with longer-term monitoring from the regional seismograph network 10

(passive seismic studies) indicated that this area is not currently seismically active. 11

Although geological studies and studies of air photos indicate that some faults in the area 12

may have been active in the geologically recent past, they do not appear to be currently 13

active. The closest measured seismic activity is just south of the international border 14

with Mexico associated with the site of the 1887 Sonora earthquake (magnitude 7.4; 15

Suter and Contreras, 2002). 16

17

The large-scale crustal structure through Clifton was determined by an active 18

seismic study that used elastic-wave energy from the open-pit copper-mine blasts at the 19

Globe, Arizona, and Tyrone, New Mexico. The line for this study was recorded with 20

seismographs along a profile extending between these mines through Clifton/Morenci. 21

Morenci is the site of a third open-pit copper mine approximately 3 km from the Clifton 22

Hot Springs. The main feature on this crustal section is a thickening of the crust from 28 1

to 32 km, over a horizontal distance of 15 to 20 km, east of Clifton/Morenci. This 2

thickening roughly coincides with the steep gravity gradient mapped by Boler et al 3

(1980), and may be related. The general nature of the seismic crustal profile does not 4

allow a direct spatial correlation with the gravity map, but crustal thickening of the 5

seismic profile is probably associated with increasing elevations to the east. An 6

additional 5.81 km/s upper layer seen in the seismic profile as the crust thickens is 7

interpreted to represent an average effect of the Datil-Mogollon volcanism (Witcher, 8

1981). What was not seen in data from the active seismic study was any evidence for an 9

anomalous crustal or upper mantle structure beneath Clifton/Morenci. This result 10

supports conclusions from the ages of volcanic rocks in the area that there is no 11

anomalous magmatic heat source associated with the Clifton geothermal system (or other 12

related geothermal systems). 13

14

The most reliable regional heat flow value published close to the Clifton 15

geothermal system is from the Bitter Creek site, approximately 35 km southeast of 16

Clifton, with a value corrected for topography of 115 mW m-2 (Roy et al, 1968). This 17

value is in the upper half of the normal range of values for southern Basin and 18

Range/adjacent Colorado Plateau (Morgan and Gosnold, 1989). Most of the heat flow 19

data collected from wells closer to Clifton have yielded values significantly different 20

from the regional norm or had temperature data indicating water flow within the depth 21

range over which the temperature data were collected (Witcher and Stone, 1980, Witcher, 22

1980). Using other heat flow data from the area (Roy et al, 1968; Reiter and Shearer, 1

1979), a conservative regional heat flow value of 100 mW m-2 is estimated for the Clifton 2

geothermal system. Such a regional heat flow is high compared with value of 50±20 mW 3

m-2 that are more characteristic of tectonically stable regions of the US (Morgan and 4

Gosnold, 1989) 5

6

In summary, apart from electrical data very close to the Clifton Hot Springs, none 7

of the geophysical techniques, including heat flow, detected any direct evidence of 8

geothermal systems in the subsurface. This result suggests that the plumbing of these 9

systems is either relatively narrow, so as to escape detection by the limited aerial 10

coverage of the geophysical studies, and/or deep so as to be undetectable against a 11

background of normal high temperatures at depth. 12

13

Modeling the Clifton Hot Springs Data 14

15

The information presented above tentatively indicates that the Clifton Hot Springs 16

are a fracture-dominated groundwater circulation system driven by the piezometric 17

gradient associated with the elevation difference between the Colorado Plateau and the 18

southern Basin and Range provinces. Data from this system are therefore suitable for 19

application of the semicircular aquifer model. 20

21

From the data presented above, the outflow temperature of the springs, TO, is 1

taken to be 70°C and the maximum aquifer temperature, Tm, 180°C. The mean annual 2

surface temperature, Ts, has been taken from the Western Regional Climate Center, 3

Desert Research Institute (http://www.wrcc.dri.edu/cgi-bin/cliMAIN.pl?azclif, accessed 4

2005-9-14) as 19°C, from which a ratio of dimensionless exit temperature, θe, to 5

dimensionless maximum aquifer temperature, θmax, of 0.32 was calculated (Equation 3). 6

Using this ratio with Figure 4, the dimensionless flow rate was determined to be 1.4, and 7

using this dimensionless flow rate, ψ, in Figure 3, the dimensionless maximum aquifer 8

temperature, θmax, was determined to be 0.95. With this information and the dimensioned 9

maximum aquifer temperature and mean annual surface temperature the undisturbed 10

maximum aquifer wall rock temperature, TR, was calculated from the relations given in 11

Equation 3: 12

13

TR = ((Tm - Ts)/θmax) + Ts, (9) 14

15

yielding a value for the undisturbed wall rock temperature of 188.5°C, which has been 16

rounded to 190°C. Using the regional heat flow value of 100 mW m-2 discussed above, 17

and assuming a thermal conductivity for crystalline basement rocks of 2.5 W m-1 K-1, a 18

regional geothermal gradient of 100/2.5 or 40°C km-1 was calculated. Assuming a 19

surface temperature of 15-20°C (most of the region surrounding Clifton is at a higher 20

elevation and therefore has a lower mean annual temperature), the depth range at which 21

the temperature is 190°C, i.e., the maximum depth of the aquifer circulation, R, has been 22

estimated to be 4.25 to 4.375 km. A rough average of this range of 4.3 km has been 1

taken for convenience. Substituting this value for R, using the previously determined 2

dimensionless flow rate, ψ, of 1.4, and assuming a thermal diffusivity, κ, of 1 mm2 s-1, in 3

Equation 6, an aquifer discharge rate of 19 x 106 mm3 s-1 or 19 l s-1 (liter per second) was 4

calculated. 5

6

The Clifton Hot Springs all currently discharge into the San Francisco River and 7

based on the change in the dissolved chloride content of the San Francisco River above 8

and below the hot springs, Witcher (1981) estimated a mean discharge of thermal waters 9

from the springs to be approximately 76 l s-1. This discharge is four times higher than the 10

discharge estimated by the semicircular aquifer model, but the agreement in magnitude of 11

the two values is taken to be strong validation for the usefulness of the aquifer model. 12

13

In terms of the disagreement by a factor of four between the semicircular aquifer 14

flow model and dissolved chloride estimate of flow, the first possible reason why the 15

semicircular aquifer model may underestimate flow is that it represents a very 16

geologically simplified groundwater flow system that is probably more efficient in terms 17

of heating water with deep penetration over a short distance of two-dimensional 18

horizontal travel. With more geologically realistic fracture flow systems, and longer 19

horizontal travel distances, and/or significant three-dimensional components to the 20

heating, larger flow rates could be heated to the same outflow temperatures. 21

22

The second reason why the semicircular aquifer model may underestimate flow in 1

the Clifton geotherm system is that it assumes a single aquifer flow system. Hot springs 2

discharge at a range of temperatures from 30 to 70°C into the river (Witcher, 1981) which 3

represent some mixing with colder waters, but also may represent the conjunction of 4

different thermal aquifer systems. The other thermal springs within a 40 km radius of 5

Clifton indicate that there are multiple thermal aquifer systems in the region. The flow 6

from only the highest temperature spring is estimated here. If flows from lower 7

temperature springs were also included, the total flow would come closer to that 8

estimated by the dissolved chloride method. 9

10

Implications for Development of the Clifton Geothermal System 11

12

If all the hot springs at Clifton are assumed to discharge at the maximum spring 13

temperature of 70°C, the maximum thermal energy that can be extracted from these 14

waters using the flow rate from the semicircular aquifer model and assuming that the 15

waters are cooled to 20°C is given by Equation 7 as 4 MW thermal. Using the higher 16

flow rate estimate from the change in chloride concentration of the San Francisco River, 17

the maximum thermal energy rises to 16 MW thermal, although the total is probably less 18

as all the hot springs are not as hot as 70°C. Using the estimates of the maximum thermal 19

energy output of the system from either the semi-circular aquifer model of based on the 20

chloride concentration estimate of discharge rate suggests that up to several MW of 21

thermal energy are available at low temperatures, 20-70°C. 22

1

If the system can be tapped at its maximum temperature indicated by groundwater 2

geochemistry of 180°C, the maximum power that can be extracted from the system is 3

estimated to range from 13 MW thermal to 51 MW thermal, using flow rate estimates 4

from the semicircular aquifer model and the chloride concentration estimate of the 5

discharge rate, respectively. If the geothermal fluid can be brought to the surface at close 6

to its maximum temperature, it could be used to generate electricity, but not all of the 7

maximum possible thermal power extraction rate from the hot springs is available to 8

generate electricity. Power is only extracted from the geothermal fluid as it cools from 9

the temperature at which it enters the turbine to the temperature at which it exits the 10

turbine, the condenser temperature. Assuming a condenser temperature of 40°C, the 11

power that could be extracted from the maximum temperature geothermal fluid is in the 12

range of 11 to 44 MW thermal. Assuming typical turbine equipment performance and 13

efficiency (66%) and generator efficiency (97%; Rafferty, 2000), the maximum electrical 14

power generation from this system for a direct steam plant would be in the range of 7 to 15

28 MW. If a binary power plant is used (heat from the geothermal fluid is transferred to 16

a lower-boiling temperature working fluid), the efficiency of the system would drop to 17

about 50% at 180°C, and would continue to decrease rapidly as the input temperature of 18

working fluid to the turbines decreases (Nichols, 1984). 19

20

Discussion 21

22

The addition of maximum aquifer temperature constraints from groundwater 1

chemistry data to the semicircular aquifer model significantly increases the information 2

that may be derived from this model. Using this maximum temperature, the outflow 3

temperature from the system, and the mean surface temperature, a solution has been 4

derived to determine the dimensionless flow rate through the system from which the 5

undisturbed wall rock temperature at the maximum depth of the system may be 6

estimated. The further addition of an estimate of the regional geothermal gradient (or 7

heat flow and thermal conductivity), then allows the maximum depth of penetration of 8

the aquifer in the system to be estimated, and ultimately the dimensioned flow rate 9

through the system. From these data the maximum thermal power that is put out by the 10

system at its outlet, and the maximum thermal power that could theoretically be tapped if 11

the full flow of the system could be tapped at its maximum temperature may be 12

estimated. These are significant results from very limited input because they allow the 13

power potential of a hot spring system to be estimated with data easily collected at very 14

low cost from the discharge waters of the springs. 15

16

Modeling hot springs as the outflow of semicircular aquifers driven by the 17

piezometric gradient is a major simplification of any natural groundwater system and 18

must be applied prudently. This model is clearly not applicable where addition of local 19

magmatic heat to the crust is a significant factor in the thermodynamics of the 20

groundwater system, or other places where thermal buoyancy forces are important in 21

driving groundwater flow. The model is also not applicable where flow is widely 22

distributed in porous aquifers, as in basin flow (e.g., the basins of the Rio Grande rift, 1

Morgan et al., 1981). However, in terrains where the aquifers are likely to be confined to 2

fracture flow and locations of hot springs indicate that they may be driven by piezometric 3

gradients, the semicircular aquifer model is probably a useful first approximation thermal 4

model to the geothermal systems. When all the approximations of the model are 5

considered, application to the Clifton Hot Springs has shown remarkably good agreement 6

between the discharge flow rate calculated from the semicircular aquifer model and the 7

flow rate estimated from the chloride budget of the San Francisco River into which the 8

springs discharge. In addition, flow rate estimates from the chloride budget may not be 9

directly comparable to the estimate from the aquifer model because the chloride budget 10

flow rate may be the aggregate of more than one aquifer system. 11

12

A primary motivation for this study has been to use data that are easily obtained 13

from hot spring systems to make a rough assessment of the energy production potential of 14

the systems. By generating a method of estimating the flow rate through the systems, the 15

thermal power output of the systems can be easily estimated. In addition, assuming that 16

the systems can be tapped at their maximum temperatures at depth, and produced at their 17

natural flow rates, a maximum thermal power output of the systems may be estimated. 18

However, these thermal power output estimates are not necessarily the power available 19

for use and the temperatures at which the geothermal fluid can be produced strongly 20

controls the applications for which it may economically be used. 21

22

With fluid temperatures below about 120°C, direct use of the thermal energy is 1

the only viable economic use because, even though electricity can be generated using 2

binary systems below this temperature, the net plant efficiency becomes very low as the 3

temperature drops (Nichols 1986) and very large fluid flow rates are required to produce 4

more than a few MW of electricity. However, direct use of the thermal energy can be 5

very efficient. As the temperature in the geothermal fluid drops the uses can cascade so 6

that a significant percentage of the total thermal power from low-temperature resources 7

may be used. 8

9

Ideally, geothermal systems would always be tapped at their maximum 10

temperature. In some situations, drilling on a thermal anomaly results in significant 11

increases in temperature with depth in the geothermal fluid from the temperatures 12

encountered at the water table (e.g., Morgan et al., 1979). However, if the semicircular 13

aquifer model is correct, although temperatures will be elevated in the system relative to 14

the undisturbed wall rock temperatures from the mid-point of the system to the discharge 15

of the system (Figure 2), fluid may only be produced from the system by tapping into the 16

fracture aquifer system at depth. The highest temperatures will only be found in the 17

system when the fluid flow is sufficiently so slow that the maximum temperature in the 18

system is not significantly lower than the maximum undisturbed wall rock temperature in 19

the system. 20

21

The semicircular aquifer model indicates that the Clifton geothermal system is 1

almost ideal in that the maximum aquifer temperature is over 90% of the maximum wall 2

rock temperature. However, drilling to over 4 km would be necessary at a very poorly 3

defined site many km from the hot springs to tap the aquifer at this temperature, and the 4

chance of missing the main fractures of the aquifer and producing a lower flow rate, 5

and/or producing fluid at a lower temperature are considered to be high. Thus, although 6

relatively high thermal power output is predicted by tapping into the maximum 7

temperature fluid of the system at its maximum flow rate, this task may not be easily 8

achieved. Existing geological and geophysical data discussed above do not indicate the 9

subsurface location of the aquifer system, nor provide indications of successful 10

exploration strategies for this task. With current techniques and understanding of the 11

Clifton geothermal system, therefore, it is likely that it could only be produced at 12

temperatures and/or flow rates significantly below those used to predict is maximum 13

potential power output, and with the current understanding of the system a conservative 14

estimate of the electrical power generation from the system would be an order of 15

magnitude less than the maximum estimate above, or within the range of 1 to 3 MW. 16

17

Concluding Remarks 18

19

A restricted class of geothermal systems has been considered here that are not 20

obviously associated with young magmatic heating or deep basin circulation and are not 21

generally represented by active exploitation in terms of electrical power generation. 22

They are numerous, however, especially in regions of Neogene tectonism. Many of them 1

have been sites of interest for electrical power generation, as was and is the situation for 2

Clifton Hot Springs in Arizona, which are discussed in this contribution. Examination of 3

these geothermal systems suggests that they are heated by circulation of groundwater to 4

depth along fracture systems, driven by a piezometric gradient associated with 5

topography, in the regional geothermal gradient. Although clearly a major simplification 6

of the geological system, the semicircular aquifer model has been developed to generate 7

estimates of flow parameters in these geothermal systems from only temperature and 8

water geochemistry temperature data. These data may be collected at the surface with 9

relative ease. The flow parameters allow calculation of the thermal power output of the 10

geothermal system at its discharge temperature, and at its maximum temperature. The 11

model has been successfully applied to the Clifton Hot Springs geothermal system. 12

13

A basic conclusion from the semicircular aquifer model is that if significantly 14

higher temperatures are required than the temperatures the system discharges, and if the 15

semicircular aquifer model is applicable, drilling to a few km will be required to tap these 16

temperatures. The location of drilling will be displaced laterally by a few to several km 17

from the discharge location. If fluid circulation is restricted to fractures, locating of these 18

fractures at depth will probably be difficult, and recovering the full fluid flow at 19

maximum temperature may require extensive regional exploration and more than one 20

drilling attempt. As these systems are likely to have a relatively modest power output, 21

typically of the order of a few tens of MW maximum, such extensive exploration and 22

drilling with uncertain returns may be economically undesirable. Tapping the geothermal 1

systems in mid-stream may also have unfavorable environmental consequences to the hot 2

spring systems downstream. However, direct use of the thermal power at or near 3

discharge of the systems may be environmentally and economically acceptable. 4

5

Acknowledgements 6

7

Jim Witcher is thanked for generously sharing materials that have been used in 8

this study and is thanked for reviewing a report that resulted in this manuscript. 9

However, any errors or misinterpretations remain the responsibility of the author. 10

11

Appendix 12

13

The solution for the dimensionless temperature, θ, as defined above in Equation 1, 14

for flow through a semicircular aquifer as shown in Figure 1 is given by Turcotte and 15

Schubert (2002, p. 265, equation 6-279) as: 16

[ ]( )21

exp(cossinζ

ζφφφζζθ+

−+−= , 17

where PerR

1148

=ζ (A-1) 18

using the nomenclature given in Figure 1, and where Pe is the Péclet number as above, a 19

dimensionless measure of the mean velocity of the flow through the aquifer which is 20

given by: 21

κruPe = , (A-2) 1

where ū is the laminar flow velocity of the fluid in the aquifer and κ is the thermal 2

diffusivity. The dimensionless exit temperature, θe, the temperature of the hot springs is 3

obtained by setting φ to π/2, the position at which the fluid discharges from the 4

semicircular aquifer. Substituting this value for φ into Equation A-1 gives: 5

[ ]( )21

)exp(1ζ

ζζθ+

−+=e . (A-3) 6

Dimensionless exit temperature may then be calculated for a range of dimensionless flow 7

rates, as shown in Figure 3. 8

9

The maximum fluid temperatures in the aquifer are determined by finding the 10

turning points of the dimensionless temperature curves shown in Figure 2. The angular 11

positions of these maximum temperature points are determined by differentiating 12

Equation A-1 with respect to angular position, setting this differential to zero (condition 13

for curve turning points), and solving the resulting equation for φ , the angular position 14

for different dimensionless flow rates. Differentiating Equating A-1 yields: 15

[ ]( )21

)exp(sincosζ

ζφζφφζζφθ

+−−+

=∂∂ (A-4) 16

Setting this equation to zero gives the result that: 17

0)exp(sincos =−−+ ζφζφφζ , or )exp(sincos ζφζφφζ −=+ (A-5) 18

There are various ways in which equation A-5 may be solved for different dimensionless 19

flow rates, for example graphical and iterative methods (e.g., Noble, 1964, ch. 2). When 20

the angular position corresponding to the maximum temperature for each dimensionless 1

flow rate has been determined, the maximum temperature θmax is determined by 2

substituting this angular position into Equation A-1. In this study Equation A-5 was 3

solved graphically in a spreadsheet program, and the maximum temperatures calculated 4

from the angular positions determined from these solutions as a function of dimensionless 5

flow rate are shown in Figure 3. 6

7

Other equations used in calculations for this study are given in the main text. 8

References Cited 1

2

Boler, F., Christopherson, K., Hoover, D., 1980. Gravity observations: Clifton-Morenci 3

area, Greenlee County, S.E. Arizona. U. S. Geol. Surv. Open File Report No. 80-4

96. 5

6

Fournier, R. O., 1977. Chemical geothermometers and mixing models for geothermal 7

systems, Geothermics. 5, 441-450. 8

9

Fournier, R. O., Rowe, J. J., 1966. Estimation of underground temperatures from silica 10

content of water from hot springs and wet steam wells. Am. J. Sci., 264, 685-697. 11

12

Fournier, R. O., Truesdell, A. H., 1973. An empirical Na-K-Ca geothermometer for 13

neutral waters. Geochem. Cosmochim. Acta, 37, 1255-1275. 14

15

Klien, D., Long, C., Christopherson, K., Boler, F., 1980. Reconnaissance geophysics in 16

the Clifton and Gillard geothermal areas, S.E. Arizona. U. S. Geol. Surv. Open 17

File Report No. 80-325. 18

19

Morgan, P., 2002. Heat Flow. In: Encyclopedia of Physical Sciences and Technology, 3rd 20

edn., v. 7. Academic Press, 265-278. 21

22

Morgan, P. W. D. Gosnold, 1989. Heat flow and thermal regimes in the continental 1

United States. In: Pakiser, L. C., Mooney, W. D. (Eds.), Geophysical Framework 2

of the Continental United States. Geol. Soc. Am., Memoir 172, Boulder Colorado, 3

493-522. 4

5

Morgan, P., Harder, V., Swanberg, C. A., Daggett, P. H. 1981. A groundwater convection 6

model for Rio Grande rift geothermal resources. Geothermal Resources Council, 7

Transactions, 5, 193-196. 8

9

Morgan, P., Swanberg, C. A., Lohse, R. L., 1979. Borehole temperature studies of the 10

Las Alturas geothermal anomaly, New Mexico. Geothermal Resources Council, 11

Transactions, 3, 469-472. 12

13

Nichols, K. E., 1986. Wellhead power plants and operating experience at Wendel Hot 14

Springs. Geothermal Resources Council, Transactions., 10, 341-346. 15

16

Noble, B., 1964. Numerical Methods, I, Iteration, Programming and Algebraic Equations. 17

Oliver & Boyd, Edinburgh & London, 156 pp. 18

19

Rafferty, K., 2000. Geothermal Power Generation, Primer on Low-temperature, Small-20

Scale Applications. Geo-Heat Center, Oregon Inst. Tech., Electronic Pubn., 21

http://geoheat.oit.edu/pdf/powergen.pdf (accessed 2005-10-25),12 pp. 22

1

Reiter, M., Shearer, C., Terrestrial heat flow in eastern Arizona, a first report. J. Geophys. 2

Res., 84, 6115-6120. 3

4

Roy, R. F., Beck, A. E., Touloukian, Y. S., 1981.Thermophysical Properties of Rocks. In: 5

Touloukian, Y. S., Judd, W. R., Roy, R. F. (Eds.), Physical Properties of Rocks 6

and Minerals. McGraw-Hill/CINDAS Series on Material Properties, v. II-2, pp 7

409-502. 8

9

Suter, M., Contreras, J., 2002. Active tectonics of northeastern Sonora, Mexico (southern 10

Basin and Range Province) and 3 May 1887 Mw 7.4 earthquake. Bull. Seism. Soc. 11

Am., 92, 581-589. 12

13

Turcotte, D. L., and Schubert, G., 2001. Geodynamics, 2nd edn. Cambridge Univ. Press, 14

Cambridge, U.K., 456 pp. 15

16

Witcher, J. C., 1979. A progress report of geothermal investigations in the Clifton area. 17

Arizona Geol. Surv. Open File Report 79-2b, 16 pp. 18

19

Witcher, J. C., 1981. Geothermal energy potential of the Lower San Francisco River 20

region, Arizona. Arizona Geol. Surv., Open File Report 81-7, 135 pp. + 3 21

enclosures. 22

1

Witcher, J. C., Stone, C., 1980. Heat flow and the thermal regime in the Clifton, Arizona 2

area. Arizona Geol. Surv., Open File Report 80-1a, 29 pp. 3

Figure Captions 1

2

Figure 1. Model semi-circular aquifer of radius R and circular cross section of radius r. 3

Distance along the aquifer is measured by the angle φ . Laminar flow velocity 4

through the aquifer is ū [Modified from Turcotte and Schubert, 2001, Figure 6-9, 5

p. 233]. 6

7

Figure 2. Dimensionless temperature θ versus angular distanceφ along a semicircular 8

aquifer in units of π radians. Dashed curve θw is temperature profile of wall rock, 9

or aquifer temperature with no flow. Solid curves represent temperature profiles 10

for different flow rates. See text for definition of dimensionless temperature and 11

flow rates. 12

13

Figure 3. Upper curve: maximum dimensionless temperature versus dimensionless flow 14

rate for semi-circular aquifer. Lower curve: exit temperature versus 15

dimensionless flow rate for semi-circular aquifer. See text for definitions of 16

parameters. 17

18

Figure 4. Plot of the ratio of exit (outflow minus mean annual surface) temperature to 19

the maximum temperature attained in the aquifer system, both in dimensionless 20

form, as a function of dimensionless flow rate. Two overlapping curves are 21

shown, the data and a 5th-order polynomial fit to the data. 22

1

Figure 5. a. Map of the United States showing the location of Arizona for detail shown in 2

b. b. Map of Arizona ahowing simplified topography, major rivers, and location 3

of the Clifton Hot Springs. [Modified from D. Minnis, Arizona Geographical 4

Alliance, Dept. Geography, Arizona State U., 5

http://alliance.la.asu.edu/maps/AZTOPO.PDF, accessed 2005-9-14]. 6

Figure 1 1

2

Figure 2 1

2

3

Figure 3 1

2

Figure 4 1

2

Figure 5a 1

2

Figure 5b 1

2