SPE

SPE 21542

Thermal Conductivity Estimation From Temperature LogsA.C. Seto and S. Bharatha, Esso Resources Canada Ltd.'SPE Member

Copyright 1991, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the International Thermal Operations Symposium held in Bakersfield, California, February 7-8, 1991.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the Society of Petroleum Engineers and are sUbject to correction by the author(s). The material, as presented, does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Societyof Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledg-ment of where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.

Abstract

Heat losses to over/underburden play an important rolein determining the efficiency of a thermal oil recoveryprocess such as cyclic steam stimulation. Thermalconductivity is a key parameter for evaluating heatlosses. Since there is generally little informationconcerning in situ thermal conductivity values, it is oftenassumed in reservoir simulation models that the reservoirand its over/underlying formations possess the samevalue of thermal conductivity. If the over/underlyingformations possess significantly higher values of thermalconductivity compared to the reservoir, the heat losseswill be underestimated in calculations based on uniformthermal properties. Systematic procedures to estimatethermal conductivity values from temperature logs andcore measurements are presented in this paper andapplied to field and laboratory data.

Introduction

The commercial process employed by Esso ResourcesCanada Limited to recover the highly viscous bitumen atCold Lake, Alberta, is cyclic steam stimulation. Theinjection of steam at high pressures (10-11 MPa, 1450-1600 psi) and temperatures (311-318 C, 592-604 OF)results in heated reservoir zones, from which heat is lostto the formations above and below the reservoir byconduction. It is necessary to estimate the in situthermal properties of the reservoir and theover/underlying formations, in order to assess the

References and illustrations at end of paper

179

thermal efficiency of the recovery process by analyticalor numerical methods. In particular, in situ thermalconductivity is a key parameter for evaluatingover/underburden heat losses. However, typically thereis little information concerning in situ thermalconductivity values and it is often assumed in reservoirsimulation models that the reservoir and its neighboringgeological formations have the same or nearly the samethermal conductivity values (see, e.g., the values used inthe simulation work by Boberg and Rotter1 andJohnson et al.2 ). If the over/underlying formationspossess significantly higher values of thermalconductivity than that of the reservoir, the heat lossesand potentially the ultimate recovery from the reservoirmay be miscalculated by assuming the same values forthe thermal conductivity of all the formations.

This paper presents a systematic procedure forestimating thermal conductivity values from temperaturelogs and laboratory measurements on cores. Anisotropyof thermal conductivity will be ignored here. Fromtemperature logs, in situ thermal conductivity ratiosbetween geological formations can be determined. Ifthe in situ thermal conductivity of one formation isknown, then the in situ thermal conductivity values ofthe entire stratigraphic column can be determined byusing the ratios. By using core measurements toestimate the thermal conductivity of one formation (saythe reservoir), the in situ thermal conductivity of thevarious formations may be determined from the log-derived ratios. This estimation procedure is illustrated byapplication to two initial temperature logs from ColdLake and data from measurements on cores from theAthabasca oil sands deposit in Alberta. A correlation

Thermal Conductivity Estimation

Let Ri == KKi denote the ratio of Ki (for any layer i) toref

the thermal conductivity Kref of a reference layer (heretaken to be the Clearwater formation). The value of Rifor each major formation above the Clearwater namelyGlacial Till. Colorado Shales. Grand Rapids (for adescription of the formations above and below theClearwater, see the article by Wightman et al. in Ref. 3)was determined from (1). using average values for thetemperature gradients of each layer. The results (Rvalues) are presented in Figs. 2 and 3.Once the ratios Ri are known, the in situ the rmaIconductivity profile of the Cold Lake stratigraphiccolumn can be determined if the thermal conductivity ofthe reference layer (here the Clearwater formation) isknown.

A literature search for thermal conductivity values of oilsands revealed that most values are obtained frommeasurements on remolded or reconstituted oil sandsamples under room temperature and atmosphericconditions. Set04 developed a transient state thermaltest cell to measure thermal conductivity and thermaldiffusivity properties of oil sand cores at temperatures,pressures and fluid saturations encountered duringthermal recovery processes. Details of the testapparatus. experimental and data analysis proceduresmay be found in Refs. 3, 4 or 5. Since every attemptwas made to ensure minimum disturbance during coringof the samples and the cores were tested undersimulated reservoir conditions, the thermal conductivityvalues obtained are expected to be representative of thein situ values. The use of a transient state thermalconductivity measurement technique also helped tominimize the effects of convection on thermalconductivity measurements, as observed in most steadystate type tests.

Table 1 lists the porosity. water saturation. oil saturation,temperature. measured and calculated (from correlationequation (4) below) thermal conductivity values for eachtest on the oil sand samples obtained from theAthabasca oil sands deposit.4 The test specimensincluded remolded and undisturbed core samples. Notethat the partially saturated, rich. remolded oil sand

i,j = 1.2.3 ,n (1)

Theory

where n denotes the number of layers considered.

Ki (dL) = KJ' (dL)dz i dz j

Tables of typical thermal conductivity values for shales.sandstones and limestones, taken from the literature. arealso included to serve as a guide for estimating thermalconductivity values when no measurements are available.

2 THERMAL CONDUCTIVITY ESTIMATION FROM TEMPERATURE LOGS SPE 21542equation for thermal conductivity as a function of However, based on the magnitude of the temperatures,porosity. fluid saturations and temperature for the it will be assumed that the temperature profiles ~ecordedAthabasca oil sands has been developed, based on core at these wells were unaffected by steaming andmeasurements. This correlation is used to estimate the in correspond to virgin reservoir conditions. The verticalsitu thermal conductivity of the Cold Lake oil sands. The solid lines showing the major formation boundaries inthermal conductivity of formations above the reservoir Figs. 2 and 3 were determined from well logs run inare then determined from the ratios obtained from logs. nearby wells. It is clear from the figures that the

average temperature gradients for the differentformations are unequal.

Application

Consider a horizontally layered model of the earth. asshown in Fig. 1. Suppose that the (isotropic) thermalconductivity of each layer is constant Ki being thethermal conductivity of layer i. For the virgin state ofthe reservoir. it is reasonable to suppose that all heattransfer is due to conduction alone. If lateraltemperature variations are ignored and steadyconditions assumed, the vertical temperature gradient isconstant for each layer -- the temperature data fromregions unaffected by thermal recovery processes areroughly consistent with the assumption of a constant

temperature gradient in each major formation. Let (~)idenote the (constant) vertical temperature gradient inlayer i. T being the temperature and z being the depth(Fig. 1). Since the vertical heat flux is the same for eachlayer

Since the temperature gradient (:). for each layer is. I

d h . Ki bknown from the temperature ata. t e ratio K' may eJ

determined for any pair of layers from (1). If the valueof Ki is known for anyone layer. the value of Kj for allthe other layers may then be determined.

The procedure will be illustrated by application totemperatu re data from two vertical observation wellsinstalled at Esso Resources' oil sands leases at ColdLake. where bitumen is recovered from the Clearwaterformation. The temperature log data are shown in Figs.2 and 3. By the time these temperature logs were run.steaming of a neighboring well pad had begun.

180

SPE 21542 A. C. SETO & S. BHARATHA 3sample was saturated with water after the first series ofthermal conductivity measurements to allow for furthertesting at a different water saturation. Similarly, waterwas used to flush through the rich and lean undisturbedoil sand specimens to reduce the bitumen saturations fortwo more series of tests. Thermal conductivitymeasurements on tailings sand (0% oil saturation)samples are also included.

Somerton et a/.6 developed a correlation for estimatingthermal conductivity values of unconsolidated oil sandsfrom porosity and water saturation. The equationpresented in Ref. 6 requires knowledge of the thermalconductivity of the solid constituent. Since thisproperty is constant when dealing with the same type ofrock, it can be incorporated into the coefficients of thecorrelation equation as follows:

where m,n,p,q are constant exponents. The results didnot improve the reduction coefficient much.

The correlation (4) will be used to estimate the thermalconductivity of the Clearwater formation. The averagevalues for porosity, water saturation, oil saturation andtemperature under virgin reservoir conditions are takento be 0.35, 0.36, 0.64 and 13C (55 OF), respectively(see Table 1 of Ref. 1 for typical values). The thermalconductivity, calculated from equation (4), is 2.07W/(m'C} (1.20 Btu/hr-ft-OF). Using the averagethermal conductivity ratios shown in Figs. 2 & 3, theaverage thermal conductivities of the Till, ColoradoShales, and Grand Rapids formations are calculated tobe 2.47, 2.07, and 2.84 W/(m'C} (1.43, 1.20, 1.64Btu/hr-ft-OF), respectively.

K = a + b + d Sw (2)

K = a + b + c VSw + d VSo + eT (3)

Somerton et a/.6 also presented a correlation that allowslinear temperature dependence of thermal conductivity.The following generalized form of (2), including a lineardependence on temperature, was used to correlate datain Table 1, for cores containing gas and oil phases inaddition to the water phase in the pore space:

where K

Swa,b,c =

thermal conductivity (W/(m C)),porosity (fraction),water saturation (fraction),constant coefficients.

It may be noted that the porosity, saturation andtemperature data required for thermal conductivityestimation are all obtainable from logs. In cases wherethermal conductivity is not known for any of thegeological formations, one may use correlations such asequation (4), that may be available in the open literature,for similar formations. Typical thermal conductivityvalues of sandstone, shale and limestone measured byvarious researchers7-22 are included in Tables 2 to 4 tofacilitate estimation in the absence of any data. It shouldbe noted that the thermal conductivity values of thematerials vary greatly depending on the fluid saturations,temperature, porosity and mineralogy. Variations inmeasurement methods and sample conditions may alsoproduce variations in the values of thermal conductivity.

Practical Implications

is the dimensionless time. In equation (7), t is the time,Kob is the thermal conductivity of over/underburden,

Due to lack of knowledge about in situ thermalconductivity values, the thermal conductivity Kob of theover/underburden is typically taken to be the same as thethermal conductivity Kr of the reservoir. The effect ofunequal thermal conductivity values for the reservoir andover/underburden on heat losses may be assessed foranalytical models of steam injection such as the Marx-Langenheim model23 . For this model, the ratio HL ofcumulative over/underburden heat loss to the cumulativeheat injected, is given by (Eqn. (5) of Ref. 24)

HL = 1 - t:[etD erfc it;; + 2V~ -1] (6)

.................................. (7)tD -where

K = a + bm + cSwn + dSoP + eTq (5)

A multiple linear regression analysis was employed todetermine the coefficients appearing in (3). . Theresulting correlation

K = 4.318 - 4.883 + 0.474VSw - 0.987VSo - 0.0024T................................................................................ (4)

(Since the gas saturation is equal to 1 minus the sum ofwater and oil saturations, it is not necessary to includegas saturation in the correlation.)

where So = oil or bitumen saturation (fraction),T = temperature (0G),d,e = constant coefficients.

fits the data quite well and has a reduction coefficient(R-squared) of 0.9696. Thermal conductivity values,calculated for the oil sand samples in Table 1 from (4),are also listed in the table for comparison with measureddata.

A more general form of (3), given below, was also usedfor correlation analyses:

181

4 THERMAL CONDUCTIVITY ESTIMATION FROM TEMPERATURE LOGS SPE21542Mob, Mr are the volumetric heat capacities of in reservoir simulators to account for changes in thermalover/underburden and reservoir respectively, and h is conductivity due to porosity, fluid saturation andthe reservoir thickness. temperature variations within the reservoir during the

thermal recovery process.

Nomenclature

Acknowledgement

The assumption of equal thermal conductivity values forreservoir and over/underburden in reservoir simulationcould lead to underestimation of over/underburden heatlosses for long term evaluation of thermal processes if,as for the Cold Lake example, the over/underburdenpossess higher values of thermal conductivity comparedto the reservoir. Use of thermal conductivity valuesestimated from the procedure outlined above is expectedto improve the accuracy of predictions from analyticaland numerical models.

For the calculations here, the value Mob = 2683kJ/(m3 . 0 C) (40 Btu/ft3 _OF) for the volumetric heatcapacity of the over/underburden will be taken from theCold Lake simulation work of Boberg and Rotter1.Adopting a value of 7x10-7 m2/s for the in situ thermaldiffusivity of the Clearwater formation, based on theestimate obtained by Vittoratos25 from analysis oftemperature data, and using the previously obtainedvalue of Kr = 2.07 W/(mC) for the thermal conductivityof Clearwater, the volumetric heat capacity of thereservoir is calculated to be Mr = 2957 kJ/(m3.0 C) (44Btu/ft3 _OF). (In Table 3 of Ref. 1, a value of 2347kJ/(m 3 . 0 C) is given for the rock volumetric heatcapacity. However, this value, corresponding to thesand grains, has to be increased for rocks containingwater. Since the true value, depending on temperatureand the nature of the pore fluids and their saturations, isactually variable in the reservoir, the constant value of2957 kJ/(m3 . 0 C) employed here appears to be areasonable approximation.) Taking the reservoirthickness h = 50 m (164 ft), plots of the heat loss ratioHL as a function of time, for various R (= KOb/Kr )values, were prepared from equations (6) and (7), asshown in Fig. 4. It is seen that for the case presented inFigs. 2 & 3, corresponding to R = 1.37, the increase inthe heat loss ratio over the normal case of equalover/underburden and reservoir thermal conductivity (R= 1.0) is about 11 % after 10 years of steam injection.For long term economic forecasts of thermal recoveryprocesses, this increase in heat loss may becomesignificant. This is particularly true in thermal processeswhere steam or hot fluid override is dominant.

The method of using initial temperature logs to estimatethermal conductivity ratios, employed here, shouldimprove the accuracy of thermal property description forreservoir and wellbore heat loss simulations.

KTz

SwSoa,b,c,d,em,n,p,qHL

tDKobKrMob

MrthRi

= Thermal Conductivity=Temperature= Depth= Porosity= Water saturation= Oil Saturation= Constant coefficients= Constant exponents= Ratio of cum. heat loss to cum. heat

injected= Dimensionless time= Thermal conductivity of over/underburden= Thermal conductivity of reservoir= Volumetric heat capacity of

over/underburden= Volumetric heat capacity of reservoir= Time= Reservoir thickness= Ratio of thermal conductivity of formation i

to that of reference formation

References

The authors wish to thank Esso Resources CanadaLimited for the permission to publish this paper. Specialthanks are due to our colleague J. M. Gronseth whoprovided the temperature data and the idea of using thisdata to define formation boundaries.

Conclusions

In situ thermal conductivity ratios of geologicalformations may be estimated from initial temperaturelogs. The estimation has been carried out usingobservation well temperature data from Cold Lake.

Thermal conductivity measurements on Athabasca oilsand cores may be satisfactorily represented by meansof a correlation equation relating the conductivity toporosity, fluid saturations and temperature. Thiscorrelation has been used to estimate the in situ thermalconductivity of Clearwater formation at Cold Lake andof the overlying formations from the ratios determinedfrom temperature logs. The correlation may also be used

1.

182

Boberg, T.C. and Rotter, M.B.: "History Match ofMultiwell Simulation Models of Cyclic SteamStimulation Process at Cold Lake," paper SPE20743 presented at the 65th Annual TechnicalConference and Exhibition of SPE, New Orleans,Sept. 23-26, 1990.

Kristiansen, J., Saxov, S., Balling, N. and Poulsen,K.: "In Situ Thermal Conductivity Measurementsof Precambrian, Paleozoic and Mesozoic Rocks onBornholm, Denmark," Geologiska Foreningens iStockholm Forhandlingar, Vol. 104, Pt. 1 (1982)49-56.

15.

16. Kristiansen, J., Saxov, S. and Balling, N.: "TheThermal Conductivity of Some Crystalline andSedimentary Rocks from Scandinavia,"Geothermal Resources Council, Trans., Vol. 6(1982) 129-132.

17. Evans, T.R.: "Thermal Properties of North SeaRocks," Log Analyst, Vol. 18, No.2 (1977) 3-12.

24. Farouq Ali, S.M.: "Heat Loss to the AdjacentFormations," Producers Monthly, Vol. 30, (May1966) 4-7.

22. Mongelli, F., Loddo, M. and Tramacere, A.:"Thermal Conductivity, Diffusivity and SpecificHeat Variation of Some Travale Field (Tuscany)Rocks versus Temperature," Tectonophysics, Vol.83 (1982) 33-43.

21. Poulsen, K.D., Saxov, S., Balling, N. andKristiansen, J.I.: "Thermal ConductivityMeasurements on Silurian Limestones from theIsland of Gotland, Sweden," GeologiskaForeningen Stockholm Forhandl, Vol. 103, Pt. 3(1981) 349-356.

18. Birch, F. and Clark, H.: "The Thermal Conductivityof Rocks and Its Dependence upon Temperatureand Composition," American Journal of Science,Vol. 238, No.8 (1940) 529-635.

19. Thomas, J. Jr., Frost, R.R. and Harvey, R.D.:"Thermal Conductivity of Carbonate Rocks,"Engineering Geology, Vol. 7, No.1 (1973) 3-12.

20. Roy, R.F., Beck, A.E. and Touloukian, Y.S.:"Thermophysical Properties of Rocks," in PhysicalProperties of Rocks and Minerals, Data series onMaterial Properties, Vol. 11-2, edited by Y.S.Touloukian and C.Y. Ho, McGraw-Hili Book Co.,New York (1981) 409-502.

23. Marx, J.W. and Langenheim, R.H.: "ReservoirHeating by Hot Fluid Injection," Trans. AIME, Vol.216, (1959) 312-315.

A.C. SETD & S.BHARATHA 5Field: Experimental Results and an ImprovedPrediction Method," Geothermics, Vol. 9,Pergamon Press Ltd., Great Britain (1980) 169-178.

Johnson, R.S., Chu, C., Mims, D.S. and Haney,K.L.: "History Matching of High- and Low-QualitySteamfloods in Kern River Field, California," paperSPE 18768 presented at the SPE CaliforniaRegional Meeting, Bakersfield, April 5-7, 1989.

2.

3. Hepler, L.G. and Hsi, C. (Eds.): A as T R ATechnical Handbook on Oil Sands, Bitumens andHeavy Oils, AOSTRA Tech. Publ. Series #6,Edmonton (1989).

4. Seto, A.C.: Thermal Testing of Oil Sands, M.Sc.thesis, University of Alberta, Edmonton (1985).

5. Scott, J.D. and Seto, A.C.: "Thermal PropertyMeasurement on Oil Sands," JCPT, Vol. 25, No.6(Nov.-Dec. 1986) 70-77.

6. Somerton, W.H., Keese, J.A. and Chu, S.L.:"Thermal Behavior of Unconsolidated Oil Sands,"SPEJ, Vol. 14 (Oct., 1974) 513-521.

SPE21542

9. Somerton, W.H. and Boozer, G.D.: "ThermalCharacteristics of Porous Rocks at ElevatedTemperatures," Trans. AIME, Vol. 219 (1960)418-422.

7. Zierfuss, H. and Van der Vliet, G.L.: "LaboratoryMeasurements of Heat Conductivity ofSedimentary Rocks," Bulletin of the AmericanAssoc. of Petro. Geology, Vol. 40 (1956) 2475-2488.

8. Somerton, W.H.: "Some Thermal Characteristicsof Porous Rocks," Petro. Trans., AIME, Vol. 213(1958) 375-378.

10. Clark, S.P. Jr. (editor): "Thermal Conductivity,"Section 21 of Handbook of Physical Constants,revised edition, Geol. Soc. of America, Inc.,Memoir 97 (1966) 459-482.

11. Cermak, V.: "Coefficient of Thermal Conductivityof Some Sediments, Its Dependence on Densityand on Water Content of Rocks," Chemie derErde, Vol. 26 (1967) 271-278.

12. Moiseyenko, U.I., Sokolova, L.S. and Istomin,V.Ye.: "Electric and Thermal Properties of Rocks,"Nat. Aero. & Space Admin., Tech. Translation No.F-671 (1972) 1-63.

13. Anand, J., Somerton, W.H. and Gomaa, E.:"Predicting Thermal Characteristics of Formationsfrom Other Known Properties," SPEJ, Vol. 13(1973) 267-273.

14. Martinez-Baez, L.F.: "Thermal Conductivity of 25. Vittoratos, E.: "Interpretation of TemperatureCore Samples from the Cerro Prieto Geothermal Profiles From the Steam-Stimulated Cold Lake

183

6 THERMAL CONDUCTIVITY ESTIMATION FROM TEMPERATURE LOGSReservoir," paper SPE 15050 presented at the56th California Regional Meeting of SPE, Oakland,April 2-4, 1986.

184

SPE21542

TABLE 1 TABLE 2Measured Thermal Conductivity Values of Samples

Thermal Conductivity of Sandstone (after Set04)Sample $ Sw So T (OC) K (W/(moC) K (W/(m'oC))Actual Calculated $ References & Remarks(fluid medium)Rich 0.440 0.038 0.675 20 1.281 1.402 Air Water Oilremolded 21 1.295 1.400 0.68-4.40 4.40-6.99 1.21-4.40 0.044-0.368 Zierluss and Van der Vliet7oil sand 49 1.231 1.332

99 1.163 1.210 0.88 2.76 1.36 0.196 Somerton8151 1.114 1.084199 0.992 0.967 0.49 1.82 1.00 0.40 ditto

0.440 0.325 0.675 20 1.734 1.580 1.13-1.38 Somerton and Boozer950 1.664 1.507 1.47-4.27 Clark10101 1.470 1.383

149 1.336 1.267 1.05-3.06 1.63-3.10 0.00-0.180 Cermak11198 1.246 1.148

2.05-2.76 Moiseyenko et al.12Lean 0.396 0.816 0.184 20 2.216 2.341 1.47-2.34 3.08-5.19 0.162-0.292 Anand et a/.13remolded 49 2.186 2.270oil sand 98 2.073 2.151 1.30-2.44 2.25-4.64 0.042-0.290 Martinez-Baez14; 56-57C

147 1.977 2.032 0.47-0.59 2.04-2.27 0.350 ditto; Unconsolidated197 1.817 1.9104.51-6.12 Kristiansen et a/.15

Medium 0.350 0.270 0.730 20 2.009 1.963 3.71-4.22 Kristiansen et al.16undisturbed 48 1.850 1.895oil sand 98 1.786 1.774

148 1.747 1.652

Rich 0.343 0.106 0.894 20 1.765 1.816undisturbed 48 1.736 1.748oil sand 98 1.659 1.626

148 1.597 1.505196 1.500 1.388

0.343 0.597 0.403 99 2.026 2.142 TABLE 3147 1.884 2.026197 1.730 1.904

Thermal Conductivity of Shale (after Set04)Lean 0.311 0.786 0.214 20 2.676 2.714undisturbed 48 2.674 2.646 K (W/(moC)) $ References & Remarksoil sand 99 2.560 2.522 (flUid medium)

148 2.394 2.403 Air Water197 2.391 2.284 1.04 1.69 Somerlon80.071

0.293 0.629 0.159 21 2.778 2.819 1.45 Somerton and Boozer949 2.767 2.75199 2.551 2.629 1.17-2.89 ClarkI 0

149 2.385 2.507198 2.477 2.388 0.87-1.04 1.21-1.38 Anand et a/.13 ; 70-250 OF

(21-121 C)Tailings 0.331 1.00 0.00 21 3.377 3.125 1.40-2.00 Evans1?sand 49 3.145 3.057

100 3.116 2.933 1.52 2.37 0.148 Martinez-Baez14; 56-57C148 2.776 2.816 1.35 1.99 0.148 ditto; 124-125 C198 2.563 2.694

TABLE 4

Thermal Conductivity of Limestone (after 5et04)

References & Remarks

Birch and Clark18; Includes temperaturedependence ot thermai conductivity up to200CZierluss and Van der Vliet?

Somerton8

Somerton and Boozer9

ClarkI 0

Cennak11

Moiseyenko et al. 12

Thomas et al.19; 40.5 CRoy et a/.20 ; Includes plols of thermalconductivity vs temperature up 10 627CPoulsen et a/.21

Kristiansen et al. 15

Kristiansen at al. 16

Mongelli et al.22 ; 20-240 C

185

R=1.36 IR=1.0

Grand I Clear-Rapids water

480400320240

ColoradoShales

16080

Till

16

6' 12C?...-O>....

::J 8ro....

0>c..E 40>I-

0I

0

'"

Layer 1, Kl

Layer 2, K 2

Layer 3, K 3

Layer i, K i

OJ

Z

Depth (m)Figure 1. Formation Layers of Different Thermal Conductivities

Figure 2. Well 1 Temperature Log

Figure 3. Well 2 Temperature Log

~N,.Vl

~to

353025105

-R=1.00~.I.jl-.-R=1.37

-o-R=1.50/+ ;;1 ---R=2.00

15 20Time (years)

Figure 4. Cum. Heat Loss / Cum. Heat Injected Vs Time

~ 60L"00>-() 500>"Cro0> 40IE::J0 30--l/)l/)0

...J

ro 200>I

E::J 100

0

16

6' 12C?...-O>....

::J 8ro.... I R=1.23 V R=1.02 I R=1.38 I R=0>a. 1.0E0> 4I- Grand IClear-

Rapids water0

I I0 80 160 240 320 400 480

Depth (m)

-00

'"

Image001Image002Image003Image004Image005Image006Image007Image008