Heat Recovery Systems & CHP Vol. 11, No. 2/3, pp. 131-139, 1991 0890-4332/91 $3.00 + .00 Printed in Great Britain Pergamon Press plc

DEVELOPMENTS IN GEOTHERMAL ENERGY IN MEXICO~PART THIRTY-THREE. SIMULTANEOUS

DETERMINATION OF THE THERMAL PROPERTIES OF GEOTHERMAL DRILL CORES

A. GARCtA, E. Co~qaatEn~,s and B. DOMt~GU~Z* Departamento de Geotermia, Instituto de Investigaciones El~ctricas, Cuernavaca, M6xico, and *Comisi6n

Federal de Electdcidad, Morelia, M6xico

(Received 10 September 1990)

Al~tract--Thermal conductivity, thermal diffusivity and specific heat capacity of five drill cores from the Los Humeros and La Primavera geothermal fields were obtained via parameter estimation. The data were obtained by fitting an analytical model to the transient teraperature ri~e caused by a line source of heat of constant strength located along the axis of the samples. The model was obtained from the solution to the problem of an infinite region bounded internally by a hollow circular cylinder with thermal contact resistance at the cylinder surface (J. H. Blackwell, Transient heat flow problems in cylindrical geometry, Ph.D. Thesis, University of Western Ontario Canada, London, Canada, 1952). Results were obtained for fused quartz and a Berea sandstone for calibration purposes and the agreement with known properties for these samples showed maximum differences of 10%. Results obtained for a dry drill core from the Los Humeros field showed a thermal conductivity of 1.36 W m -~ K -I, a diffusivity of 0.54 × l0 s m ~ s -~ and a specific heat capacity of 0.961dkg -I K -~. Results for dry drill cores from the La Primavera geothermal field showed thermal conductivities between 1.53 and 2.51 W m-~ K-~ while thermal diffusivity varied from 0.71 to 1.0 × 10 -s m 2 s -~. Specific heat capacity varied from 0.73 to 1.03 kJ kg -~ K-~. Results obtained for these cores under water saturation conditions showed significant increases in all three properties with the degree of increase being a function of the pore volume occupied by the water.

a prove internal radius [m] b sample internal radius [m] C parameter C o specific heat capacity [kJ kg- ~ K- ~] H thermal conductance [W m -2 K -t ] K thermal conductivity [W m - ~ K- ~] L sample length [m] Q supplied heat [W m -~] r radial coordinate [m] SSQ sum of squares function [K 2] T temperature [K] t time [s]

Subscripts

e

ef i P r

$

w

1 2

experimental effective internal predicted rock solid water refers to probe refers to infinite medium

Greek letters

~ thermal diffusivity [m 2 s- ~] ~ probe constant y Euler's constant (dimensionless) p density [kg m -~] pC o volumetric heat capacity [kJ m -a K -~] ~b porosity [%] Z summation (dimensionless)

N O M E N C L A T U R E

131

132 A. GARCiA et al.

INTRODUCTION

Engineering studies for the development of geothermal resources focus on achieving an optimum recovery of their thermal energy. Questions such as a resource's size, type of reservoir boundaries, heat and mass reserves and deliverability, etc., are tackled through the application of reservoir engineering. These studies, however, require a knowledge of the thermophysical properties of the reservoir's fluids and rocks in order to obtain realistic answers to the questions posed. Rock properties are normally obtained by the core analysis technique in which a piece of rock core is cut during well drilling at a depth considered important to extract useful information from the reservoir. The core is subjected to in-situ conditions in the laboratory and its properties are determined. Since core extraction is expensive and full recovery of cores is not normally achieved one must be able to obtain as much information as possible from the sampled rocks.

Although steady-state methods provide accurate thermal property results, and are based on simple theoretical models, equipment design requires considerable experience to do the job. Long test periods and relatively large samples are required and water migration occurs in humid samples which leads to large errors in the determinations. On the other hand, transient experiments require a detailed mathematical analysis but the equipment and the test are simple [2] and it takes only a few minutes to complete one run. One such case of transient heat flow experiments is the transient temperature rise of an infinite medium caused by a line source of heat which, in turn, is determined by its thermal conductivity and diffusivity. Useful thermal property information for the infinite medium can be obtained from data generated in a single experiment by fitting an appropriate mathematical model which describes the physics of the experiment. That is, by fitting such a model to obtain its parameters from an agreement between experimental and model values. Such a procedure is called parameter estimation.

Model parameters are obtained by least-squares for either linear and non-linear cases depending upon whether the model is linear or not in its parameters [3]. Parameter estimation is an extensively used numerical technique and was proposed as early as 1954 [4] for the determination of insulating material properties using the procedure developed by Blackwell [1] for the determination of thermal properties of mason's sand. Beck [5, 6, 7] and Beck and Dhanak [8] described the design of experiments and the determination of thermal conductivity and diffusivity and specific heat capacity by non-linear parameter estimation of re-entry and rocket-nozzle materials and metals. Sasaki et al. [9] determined the thermal diffusivity of soils employing the non-linear parameter estimation technique. Rooke and Taylor [10] applied the technique to the determination of the thermal diffusivity of fibrous insulation, while Sartori and Francis [11] described a method based on parameter estimation to determine the thermal conductivity and diffusivity of rocks. Yankelev and Guseva [12] applied the technique to the determination of thermal conductivity and volumetric heat capacity of insulating materials. Similar procedures for obtaining the hydraulic diffusivity and permeability of reservoirs are extensively used in the analysis of well tests [13, 14].

In this work the parameter estimation technique was employed to determine the thermal properties of seven samples. First, quartz and a Berea sandstone of known properties were used to test the procedure. Then, an andesite from the Los Humeros Mexican geothermal field was analysed and the results are compared with the properties determined by other methods [15, 16]. Finally, the technique was applied to the determination of the thermal properties of four drill cores from the La Primavera Mexican geothermal field [17].

ANALYSIS

The objective of parameter estimation is to find values of the unknown parameters of a mathematical model that is fitted to experimental data. The criterion that is satisfied can be described as the model "best" parameter estimates that yield a minimum of the sum of square differences between model and experimental data [18]. For determining thermal properties from transient temperature measurements, this is achieved through the minimization of the sum of squares function:

SSQ = ~ (Tp - Te) 2 (I) 1

Developments in geothermal energy in Mexico~XXXIII 133

where:

Tp = temperature predicted by the model To = measured temperature n = number of measurements.

Temperatures Tp may be calculated by means of an algebraic model in explicit form [3] or by numerically solving the appropriate heat conduction problem [5]. If the model is non-linear, minimization of the sum of squares function requires a non-linear regression procedure which is iterative by nature.

Theoretical model

The theoretical model is obtained from the solution [1] of the heat equation for an infinite region, bounded internally by a hollow circular cylinder with thermal contact resistance at the outer cylinder surface. The problem is specified as follows:

O=T~ I OT~ I OT~ Or ~ + . . . . (2) r Or ~ Ot

T I = T~; t = 0 ; a < r < ~ (3)

K2 ~ T2 - ~ = H ( T ~ - T 2 ) ; r = b ; t > 0 (4)

aT~ K2 ~ (2~b) = ~ / L - Mt C~ ~ TI " ~t ' r = b ; t > 0 (5)

~ T 2 a ~ ' Ol C~ (32 -- a2) ~Ti . - K ~ - - = - - , r = b ; t > 0 (6)

~r b 2b ~t

a ~ ' fi ~Tt. - b 2 ~t ' r = b ; t > 0

T 2 ~ 0 w h e n r ~ ~ ; t > 0 .

The solution [1] to this problem resistance. It is given by:

where:

(7)

(8)

or as:

in which:

is very close to the line heat source solution for no contact

Tl ( t ) = A In(t) + (B) + (1 / t ) [C In(t) + D] (9)

A = Q / 4 n K L

B = A [ln(4~/b 2) _ ~ + 2K/bH]

C = AEb2/2~

D = (Ab2/2~)[E ln(4~/b ~) - V + 1 + 2K/bH]

E = 1 - ~6/bK.

For sufficiently long times, the terms of the order ( l / t ) are negligible and the variation of Tt against t is linear. Furthermore, C and D vary with b 2 and can be neglected if the probe has a small radius. Thus, the expression for Tt may be written as:

Tl = ( Q / 4 n K L ) [ l n ( 4 ~ t / b 2) - y + 2K/bH] (10)

T~ = Cl [ln(C2t) - 7 + C3] (11)

Ci = A

C2 = In (4~/b 2)

C3 = 2K/bH.

134 A. GARCiA et al.

Therefore, if H-~ o~ (C3-~0), i.e. perfect contact, this equation reduces to the solution of a simpler problem [19], given by:

T(r , t) = C~ [In(C2 t) - y]. (12)

By comparing equation (11) with equation (12) it is noted that the experimental and theoretical temperatures will differ by a constant amount in the region where they apply. That is, they differ by an amount equal to the product C~ C3 which implies that the two curves will be parallel in this region. Thus, parameter estimation can be used to fit Blackwell's model, equation (12), to the experimental data in such a region. Such a model is therefore used in the sum of squares function, equation (1), to predict the temperatures Tp. Once the best parameters of the model are obtained, thermal conductivity, thermal diffusivity and thermal contact resistance follow immediately from C~, C2 and C3. Then, volumetric heat capacity and specific heat capacity follow from the definition of thermal diffusivity and sample density, respectively:

~t = K / p C p . (13)

The estimation of the model parameters is optimized using the Gauss-Newton algorithm [18]. In this method, the analytical model is linearized by expanding it in a Taylor seres. Convergence to final parameter values is fast if the initial estimate of the parameter vector is near the minimum sum of squares, i.e. within +_ 30% of converged values [5]. This can be achieved with experience.

EXPERIMENTAL ASPECTS

Transient heat transfer experiments were conducted in the test system shown in Fig. 1. It was specifically designed for the determination of rock properties subjected to in-situ conditions. Samples may be subjected to temperatures as high as 400°C and pressures as high as 4.5 MN. Its capacity permits the determination of thermal, mechanical and fluid flow rock properties. Details about this system can be found elsewhere [20]. The measurements were carried out at room temperature using cylindrical samples dried for 24 hours in an oven with automatic temperature control. For tests under water saturation conditions, the samples were saturated with distilled water after subjecting the samples to vacuum drying for one hour. Nominal sample diameters were 5.08 cm (2") or 10.16cm (4"), depending on availability, and a length of 10.16cm (4") was used

I pressure intensifier

confining fluid

pressure intensifier

~pore fluid

upper

presst

ro

vess~

SV = servovalve

Fig. I. Schematic diagram of rock test system.

ty

re

connection ponel

I i A/D

converter

data logging system

Developments in geothermal energy in Mexico--XXXIII 135

in all cases. A longitudinal borehole 0.48 cm (3/16") in diameter drilled along the sample axis was used to accommodate the thermal probe with a tight fit.

The probes were constructed with a commercial Nichrome heater, housed in an Inconel tube, to which a type-K thermocouple was welded on the outside. In some tests, the heater-thermoeouple assembly was located inside a thin-wall bronze tube 0.48 crn in diameter and the voids left in the bronze tube were filled with copper powder to optimize heat transfer. In other cases, the heater-thermocouple assembly was inserted in the sample borehole and the voids between sample and heater were again filled with copper powder. Once the probes were located in the samples, a piece of ceramic insulator and metallic endcap were attached on one extreme of the sample to provide thermal insulation and avoid longitudinal heat losses.

A fully instrumented sample is shown in Fig. 2. As seen there, some of the samples were also instrumented with observation thermocouples in boreholes drilled at a known distance from the centre. Similar thermocouples were attached to the sample surface to monitor the temperature at that position. Finally, another ceramic insulator-metallic endcap assembly was attached to the other extreme of the sample. To start a test, the sample was located on the mount base located on the pressure vessel closure of the test system and all electrical connections were made and carefully monitored power (voltage and current) was then applied using a de power supply. Electrical test parameters were read using a digital multimeter. The temperature rise resulting from the constant heat generation per unit length in the probe was recorded using a data-logging system at a frequency of three readings per second. Typical heating times varied from 3 to 5 min, depending upon the supplied power density.

O) top view ~ type- K ~ healer-thermocouple

thermocouples combination probe

~- 5.08 cm -t ,oo ooo b) cross-section (Scm thick)

view ~ ~ ~ (4.5 cm ceramic insulation thick'

nichrome heater

type fhermocouples

_ _ test sample (10.16 cm long)

sample borehole (0.48 cm diameter)

heater- thermocouple combination probe

l i r a Rm to data'

logging system to dc power supply

- - to multimeter

Fig. 2. Two views of an instrumented sample.

HRS I 1 / 2 / ~

136 A. GARCiA et al.

Table 1. Experimental thermal property data for quartz, Berea sandstone and a drill core from the Los Humeros field, Mexico

Thermal properties K :t H

Sample Position, theory used Probe type ( W m - t K ~) (106m"s -~) ( W m - 2 K ~)

Fused quartz Internal, line source 1 1.33 - - - - Paramater estimation 1 1,32 0.86 959

Fused quartz Internal, line source 2 t.38 - - - - Parameter estimation 2 1.38 0.85 452

Berea sandstone Internal, line source 1 2.19 - - - - Parameter estimation 1 2.18 1.13 425

Berea sandstone External,* parameter estimation 2 2,10 1.23 - - Jaeger 2 2.13 1.26 - -

Andesite from Los humeros Internal, parameter estimation 2 1.36 0,54 422 External,* Jaeger 2 1.40 0.54

*Data obtained with observation thermocouple, 1.26 cm from the center.

RESULTS AND DISCUSSION

Thirteen experiments were carried out altogether. They include four runs using fused quartz and a Berea sandstone. These runs allowed calibration of the experimental equipment and of the procedure. The remaining experiments were performed using drill cores from the Los Humeros and the La Primavera geothermal fields, Mexico. Dry and water saturation conditions were employed in the samples from the latter field. For the andesitic core from the Los Humeros field and the sandstone, temperature was also measured at a point 1.26 cm from the centre to permit sample properties determination by other data reduction methods. The results are shown in Tables 1 and 2.

Experiments on quartz and Berea sandstone

For fused quartz, the two probes described above were used. The data were analyzed with standard line source theory (see for instance [21]) and using the parameter estimation technique to fit equation (11) as already described. Figures 3 and 4 show the variation temperature with time and with In (t), respectively, for a typical experiment in fused quartz. The results obtained from the two experiments are in close agreement with sample manufacturer data: 1.38 W m -t K -t, ~ = 0.83 x 10 -6 m 2 s -~. From these results, it was found that the two probe types reproduce sample properties quite well and, since the sample properties are known, the estimation technique can be used advantageously to obtain thermal contact resistance ( l /H) values at the heater-sample interface. It is seen that for the first (type-l) probe, the contact is better than that for the experiment with the other (type-2) probe. This finding is altogether expected since the outer tube of the probe provides a better and more uniform surface for contact. The thermal contact resistance values thus found are in good agreement with those reported by Cull [22].

Using the probe temperature for the Berea sandstone, thermal conductivity values of 2 . 1 9 W m - l K -t and 2 .18Wm-~K -t were found by line source theory and by parameter estimation, respectively. The thermal diffusivity result for this sample was 1.13 × 10-6 m 2 s-t using the latter technique. Using the temperature measured away from the centre, a thermal conductivity of 2.10Wm -t K -1 was found by parameter estimation and of 2.13 W m -t K -~ by Jaeger's [23] reduction algorithm. The respective thermal diffusivity results were 1.23 and 1.26 × 1 0 - 6 m 2 s -~. A thermal conductance of 422 W m-2 K-t was found for this experiment. These sets of results agree

Table 2. Experimental thermal property results from drill cores from the La Primavera geothermal field, Mexico

~t K 106 pCp Cp

Core Condition (Win I K ~) (m2s -~) ( k J m - ~ K -~) ( k Jkg ~K -m)

PR2-5A Dry 1.53 0.71 2151.5 0.97 Saturated 2.37 0.92 2570.0 1.25

PRI I-2B Dry 1.98 1.00 1980.0 0.73 Saturated 2.77 1.06 2604.9 1.00

PRI2-2A Dry 2.51 0.92 2726.1 1.03 Saturated 2.97 1.06 2793.0 1.09

PRI3-2 Dry 1.85 0.82 2247.6 0.88 Saturated 2.34 1.11 2749.3 1.11

Developments in geothermal energy in M e x i c o - - X X X I I I 137

50

40

o... 5O

. _

~. zo r:

10

~ (line source theory)

somple: fused quortz

o l I I I I I I 0 50 60 90 120 150 180 210

Time (s) Fig. 3. Variation o f the experimental temperature rise as a function of time for a typical run in fused

quartz.

to within _+ 8% and are in the expected range for such material, although, a direct comparison with literature is not possible since published data vary considerably from source to source.

For the drill core from the Los Humeros field, the temperature measured at r = 1.26 cm was analyzed by parameter estimation and by Jaeger's [23] algorithm. Thermal conductivity results were 1.36Win-inK -m, and 1 .40Wm-~K -~, respectively, while a thermal diffusivity result of 0.54 × 10 -6 m 2 S -~ was found by both methods. It is readily noted that the two sets of results are in quite good agreement as is the value of thermal conductance at the probe-sample interface with those obtained for the quartz and the Berea sandstone experiments. Comparison of results with ones typical of other andesites from the same geothermal field [16] indicates that the present determinations agree quite well and fall very close to the average values of conductivity for the igneous rocks from the Lox Azufres geothermal field, Mexico [21].

50 -

4O meo

o~

~ 3o ' E

o~

-~ / J ~ theoreticol rise ~'E 20 ~ ( fine ~ource ~heory)

~ somple :

o ~ I I I I I I 0 1 2 3 4 5 6

~og e (~)

~i 8. 4. YaHatio~ of th~ ~x~enta l ~ a t u r ~ H~ as a func~on of 10~ (~) fo~ a typic! ~a in f u ~ quartz.

138 A. GA~tCiA et al.

Results f rom experiments on drill cores f rom the La Primavera f ie ld

Table 2 contains the thermal conductivity, thermal diffusivity and heat capacity data obtained from dry and water saturated experiments by parameter estimation. Conductivity and diffusivity were obtained from the respective model parameters while volumetric heat capacity was obtained from the definition of thermal diffusivity (~ = K/pUp) . Specific heat capacity then followed from the volumetric heat capacity and measured [24] bulk density. Thus,

Cp = pCp/Pr~ (14)

where:

P~w = Pwq~ef + ps(1 - ~ber).

Table 2 shows a variation of thermal conductivity from 1.53 W m -1K -1 to 2.51 W m -~ K -~ for the dry cores and from 2.34 Wm -1 K -1 to 2.97 W m -1 K -~ for the saturated samples. It is se~n that sample PR-12 has an unusually high thermal conductivity but it also exhibits the least increase in conductivity upon saturation. This is due to the fact that it also has the lowest porosity of this s~t of samples, and therefore accepts less water. Thermal diffusivity varied b~tween 0.71 x 10 -6 m 2 s -t and 1.00 x 10-rm:s -~ under dry conditions and from 0.92 x 10-rm~s -~ to 1.11 x 10-rm:s - | when saturated. The resulting specific heat capacity falls in the range from 0.73 kJkg -~ K -~ to 1.03kJkg-~K -t for the dry cores while its variation goes from 1.00kJkg- tK -1 to 1.25 kJ kg -1 K -t upon saturating the samples with water. The conductivity and diffusivity data shown in Table 2 fall within the range of values obtained on dry drill cores from the other two geothermal fields from the Mexican Volcanic Belt [16, 21] as does the specific heat capacity data. It is also seen that all three properties increase when the samples are water saturated. This fact is altogether expected since the water fills the voids that had previously been filled with air and is in direct relation with the effective porosity of the sample. The properties determined by this methodology were cross-checked by parameter estimation using the Marquardt algorithm [18]. The differences found were less than _+ 3% for the dry samples and less than + 10% for the saturated cores. Thermal conductivity was also compared with independent results obtained using the classical line source theory. The comparison showed maximum differences of _+ 5%. Finally, the volumetric heat capacities were compared with values predicted by Kopp's Law:

pup = ps C~(1 - ~bef) + pwCpwq~ef. (15)

The differences were less than 10%.

CONCLUSIONS

Thermal conductivity and diffusivity and heat capacity data were obtained for a number of drill cores from the Mexican geothermal fields of La Primavera and Los Humeros using the technique known as parameter estimation. Results were also obtained for fused quartz for calibration purposes and for a Berea sandstone. Results for quartz were in excellent agreement with known values while results for the Berea sandstone obtained by parameter estimation agreed closely with values obtained by Jaeger's algorithm. Comparison among the various sets of results was within • --8% and comparison with data from the literature showed satisfactory agreement.

Results obtained for the sample from the Los Humeros field by parameter estimation were in close agreement with those obtained by Jaeger's algorithm. The results obtained on dry and saturated cores from the La Primavera geothermal field were in excellent agreement with data for other igneous rocks of geothermal fields from the same Volcanic Belt. The value of these properties increases when the measurements are made under saturation. This increase is directly related to the sample effective porosity. Thermal conductivity was compared with standard line source measurements and differences were less than _ 5%. Volumetric heat capacity was compared with data obtained from Kopp's Law. Differences found were 10% at most. Estimation of the model parameters by Marquardt's algorithm produced sample properties which differed by no more than 10% for all cases. The parameter estimation t~chnique is powerful and allows simultaneous determination of thermal properties from a single experiment.

Developments in geothermal energy in Mexico~XXXIII 139

Acknowledgements--Thanks are due to Dr David Nieva for his support for the publication of this work. Thanks are also due to staff members from the La Primavera geothermal field for providing the samples for this study. Mr Adrian Patifio and Mr Ricardo Oliver helped in producing the final manuscript.

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