Thermal conductivity models fro two-phase systems

Phys. Chem. Earth, Vol. 23, No. 3, pp. 351-355, 1998 © 1998 Published by Elsevier Science Ltd. All rights reserved Primed in Great Britain 0079-1946/98 $19.00 + 0.00

Pergamon

PII: S0079-1946(98)00036-6

Thermal Conductivity Models fro Two-Phase Systems B. Troschke and H. Burkhardt T e c h n i s c h e Universit~it Berlin, F a c h g e b i e t A n g e w a n d t e G e o p h y s i k , Sekr. A C K 2, Ackerstr. 71-76, D-13355 Berlin, G e r m a n y

Received 25 April 1997; revised 8 September 1997; accepted 9 December 1997

Abstract. Measured values of thermal conductivity from crystalline and sedimentary rocks are compared with values calculated from measurements on cuttings. Various models have been suggested to describe the thermal conductivity of n-phase systems. The geometric mean model, the layer model and the dispersion model are most practical because no structural information is required. These models are tested for crystalline KTB rocks and sedimentary rocks from the north-east part of the German sedimentary basin. Cores were measured intact, as well as ground to cuttings and the matrix conductivity was determined from measurements on fragment-water mixture (twophase system). Different structural characteristics of these systems require different models. However, in general and despite the lack of a physical argument, the geometric mean model provides the best agreement between measured and calculated values for rocks with low porosity (e.g. crystalline rocks). For sedimentary rocks with high porosity and high contrast between the thermal conductivity of the rock matrix and water thermal conductivity a good agreement for calculated and measured rock thermal conductivity was attained using a combination of layer and geometric mean model. © 1998 Published by Elsevier Science Ltd.

Several models were tested on a large number (81) of cores provided by the German Continental Deep Drilling Project (KTB). The KTB-driU section is composed of paragneisses and metabasites with some rare intercalations of hornblende gneisses, postmetamorphic lamprophyric and aplitic dikes. The values of rock thermal conductivity and the values from fragment measurements for crystalline rocks are taken from Pribnow (1994). In association with the Project "Evaluation of geologic and economic conditions for utilizing low-enthalpy hydrogeothermal resources" at the GeoForschungsZentrum Potsdam (GFZ) it was possible to examine sedimentary rock cores from three hydrogeotherrnal wells located in the north-east part of the German sedimentary basin. The rock collection contains 15 high and low consolidated sand-, one clay- and three marl-claystones.

2 Two-Phase Mixing Models Mathematical models of the fragment-water mixture for the evaluation of the rock thermal conductivity from the values measured on the mixture have been suggested and applied by several authors. The dispersion model (Hashin & Shtrikman, 1962) is based upon the model for magnetic permeability described by Maxwell. Spheres with thermal conductivity ~-1 are embedded in a medium with thermal conductivity k2. This model contains two limiting cases. A high conductivity case (upper limit; ~ ) for spheres with low conductivity embedded in a medium with high conductivity and a low conductivity case (lower limit; ~.L) when the spheres have the higher conductivity and the medium the lower. For measurements on a two-phase system the lower conductivity is given by the thermal conductivity of water (kw = 0.6 W m l K a) and the higher by the thermal conductivity of the rock matrix kM.

1 Introduction Knowledge of the thermal conductivity (~.) of rocks is necessary for the calculation of heat flow or for the longtime modelling of geothermal resources. Due to high costs in-situ measurements of the thermal conductivity or core recovery in many wells are strongly restricted. To determine thermal conductivity in these cases k can be estimated either from measurements on cuttings or from the mineralogical composition. Certainly all influences of the texture (anisotropy, grain bond) and of the characteristics of the porespace (porosity, internal surface, saturation, permeability) are lost with measurements on cuttings, but by using a suitable rock model it is possible to calculate a mean rock thermal conductivity. 351

B. Troschke and H. Burkhardt

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More realistic values are provided by taking the arithmetic mean of these extremes (Horai, 1991), which describe a parallel arrangement of kp~rand h r .

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where ~ is the volume fraction of water in the mixture. Taking into consideration that both limits give the extreme models for a mixture of fragments and water, Horai (1971) used the arithmetic mean of these extreme values to describe the two-phase system more realistically. Z,L+U = ~'L + Z u 2

(2)

The layer model assumes a layered arrangement of rock matrix and water with layer thickness proportional to the volume fraction of the two phases (Birch and Clark, 1940). The layered model considers heat flow parallel and perpendicular to the layer, and once more yields an upper and a lower limit case of thermal conductivity. The parallel component yields the upper limit of thermal conductivity (~a~) and the perpendicular component the lower limit

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Beck and Beck (1965) favour the geometric mean of ~ r and ~p~, because this describes a mixture of randomly distributed, equal amounts of parallel and series arrangements.

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(5)

Another model that has been suggested by several authors (Woodside & Messmer, 1961; Sass et al., 1971, Pribnow et al., 1995) uses the geometric mean of both components. Although there is no physical basis for this model, good results are obtained. ~'c = ~)~*L~w

(6)

Models which contain information about the grain shape, like the spherical porespace model (Zimmerman, 1989), were also tested. The test of the dispersion model and the spherical porespace model and the results are discussed for crystalline rocks by Pribnow (1994) and Pribnow & Sass (1995) and for sedimentary rocks by Troschke & Burkhardt (1997).

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3 Comparison of conductivities from crystalline rock cores and fragments

~w A half-space line source (HLS) and a divided bar (DB) were

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Fig. 1. Thermalconductivityof two-phasesystemsas a functionof water fraction.Calculatedvaluesfrom geometricmean model(Geo;equation(6)) and layermodel(Lay,equation(5)) usingaveragecore valuesfor fragmentconductivityof gneisses(left) and amphibolites(right).

Thermal Conductivity Models for Two-Phase Systems

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Fig. 2. Valuescalculatedfrom fragmentmeasurementsvs. measuredvaluesfrom cores with linearregression(solid lines).The comparisonincludeall measured crystallinerocks (gneisses and amphibolites)and both methods (LS and DB). Geometricmean model (GEO, left), Layer model (LAY, right) used to measure the thermal conductivity of intact rock cores. Both methods were also used to measure the conductivity of fragments from crushed sections of cores used in this comparison. The size of the fragments is comparable with cutting material. For rocks with very low porosity (4 < 0.01) the thermal conductivity of the rock matrix ( ~ ) is nearly equal to the rock thermal conductivity. Measurements and interpretation of the data were executed by Pribnow (1994). Figure 1 shows the relationship between measured thermal conductivity for two-phase systems and the fraction of water. It displays the difference in compaction of the mixture for the two methods. For the DB the compaction is generally lower and the difference in compaction for gneisses is not as high as for the amphibolites. Theoretical values for the layer model (equation (5)) and the geometric mean model (equation (6)) have been calculated using the average value of the rock thermal conductivity for both rock types (~'Amph. = 2.52 W/mK; ~,ci,s = 3.46 W/mK) as ~-MDefiance of the variation of the measured values the theoretical values for the geometric mean coincide better with measured values than the layer model, especially for higher rock thermal conductivity and greater range of water fraction (Fig. 1 right). For a more exact examination, equation (5) (layer model) and equation (6) (geometric mean model) have been solved for ~.M. Regarding the measured values of the two-phase systems as ~.p~.~ and h3 in the concered equation, ~.M can be calculated for each discrete measurement. A comparison of values calculated from fragments against measured values from cores (X¢o=) shows (Fig. 2 right) that the regression line for the layer model is an overvaluation for this model. The regression line increase slightly with increasing contrast XM/XW,anyway this can be considered as

a parallel postponement of the regression line over the given range of the rock thermal conductivity. One reason for this could be that not all the water in the mixture is effective for the measurement and the difference between the regression line and the one to one correspondence arises from an overdetermination of the water fraction (e.g. not definable portion of water on the side surface of the LS or on the cell walls of the DB). By lowering the average fraction of water (4) the calculated values of XM can be fitted to ~o= (Pribnow and Sass, 1995). A comparison of ~, calculated from the geometric mean model with measurements taken from intact cores (Fig. 2 left) demonstrates good agreement over the given range of contrast ~M/XWand the given scattering of the rock thermal conductivity values. The average values for all cores (~o,~ = 3.09 W m I K t ) and all calculated XM (XM = 3.06 Wm-lKt ) can be considered as equal. Although all influences of the texture were lost, the geometric mean model delivers for all tested models the best characterization of mean rock thermal conductivity for crystalline rocks. The test of other models, effects of anisotropy and the results are discussed more in detail by Pribnow (1994) and Pribnow & Sass (1995).

4 Comparison of conducdvities from sedimentary rock cores and fragments

The thermal conductivity of the sedimentary rocks were measured by the line source method. In order to measure the thermal conductivity of cores a half-space line source and in the case of the two-phase system, a full-space line source (needle probe) were used. After the measurements were taken on the intact cores, the pore volume was determined

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Fig. 3. Values measured from cores (full symbols) and two-phase systems (open symbols) vs. water fraction (GEO: geometric mean model; Lay: layer model) and then the rocks were crushed. In contrast the measurements outlined in chapter 3 the rocks were ground to grain size. Therefore all dependence on grain bond and pore space are lost. The two-phase system contains all components of the milled rock inclusive the cement clays of the sandstones. There is no compaction by measuring the thermal conductivity of a grain-water mixture by a needle probe. Therefore for a two-phase system a minimum fraction of water is necessary. For a water fraction lower than 28 % the system can be described as a three-phase system (air-water-grain). All models were used to determine ~.~ and by using the known pore volume were also used to calculate the rock thermal conductivity. Only for sedimentary rocks with low contrast between matrix thermal conductivity ~.~ and the thermal conductivity of water, is it possible to describe LM and ~or~ theoretically with one model. In this case the best agreement is also given by the geometric mean model. For sandstones this ratio between the thermal conductivity of water and the matrix thermal conductivity is generally greater than five. In this case no single model can be found that can estimate the rock thermal conductivity from the calculated matrix thermal conductivity. Therefore for the two different systems, grain-water and rock-porefluid, a model combination is necessary to estimate the rock thermal conductivity. Model combination means that in a first step the thermal conductivity of the rock matrix (kM) will be determined with one model and in the second step the thermal conductivity of the rock will be calculated with another model. Figure 3 shows the relationship between k for two datasets (one measured on cores and the other on the cutting-water mixture) against the fraction of water. Theoretical values for each rock type have been calculated using the layer model (equation (4); dashed line) using the average value of from two-phase system measurements and the fraction of water to estimate ~-M- With the geometric mean model (equation (5); solid line) theoretical values for rock thermal

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Fig. 4. Comparison of rock thermal conductivity calculated from cutting measurements (full-space LS) by model combination vs measured from cores (half-space LS). The comparison include all measm,ed sedimentary rocks. Dashed lines are ±10% deviation from one to one correspondence(solidline) conductivity can be calculated by using XM and the known porosity of the cores. Of course there is no good agreement between the theoretical values and the measured values of the sandstones, because the fluctuation of the thermal conductivity of the rocks is higher than 1.9 W/mK. A more detailed assessment is given by a comparison for each discrete measurement. Comparison of the calculated rock conductivity using the model combination (the layer model to calculated XM and the geometric mean to calculated the rock thermal conductivity) with the rock thermal conductivity measured on cores, shows good agreement (Fig. 4). The deviation for single values is in general lower than 10 % and despite the scatter, the average value of the cores measurements and the calculated values are nearly equal. This indicates that a model combination could be used to correct the influence of high porosity and the effects of a high contrast between the thermal conductivity of water and thermal conductivity of the rock matrix.

5 Summary and conclusion

Different methods of estimating thermal conductivity of rocks from fragment measurements were tested. For a fragment water mixture the geometric mean model seems to be the most satisfactory. All models were tested using a water fragment mixture and water saturated cores. Therefore the results are only valid for this case. For crystalline rocks with very low porosity the calculated thermal conductivity XF is almost identical to the rock thermal conductivity. Therefore an average rock thermal conductivity can be directly estimated using the geometric mean model. However this model provides no information about anisotropy. These observations confirm the results of other comparisons of crystalline rocks (Sass et al., 1971 and 1992)

T h e r m a l C o n d u c t i v i t y M o d e l s for T w o - P h a s e S y s t e m s For s e d i m e n t a r y rocks with high matrix conductivity the estimated thermal conductivity m u s t be corrected for the effects o f porosity. T h e r e f o r e c o m b i n a t i o n s o f m o d e l s were tested to describe the two different s y s t e m s (water-grain and intact core). A g o o d a g r e e m e n t for calculated and m e a s u r e d rock thermal conductivity was attained u s i n g a c o m b i n a t i o n of layer and geometric m e a n model. T h e layer m o d e l to calculated LM u s i n g X m e a s u r e d on the t w o - p h a s e s y s t e m and the g e o m e t r i c m e a n m o d e l to calculated the rock thermal conductivity u s i n g the calculated kM and the porosity o f the rocks. For s e d i m e n t a r y rocks with a low contrast b e t w e e n the thermal conductivity o f the rock matrix and water thermal conductivity, the rock thermal conductivity can also be calculated u s i n g only the geometric m e a n model. Acknowledgements: We thank gratefully Daniel Pribnow who provided his data and results for this article. We are grateful to KTB and GFZ for there support. We are also grateful to nmo Kukkonen and Ulf Seipold as the reviewers of this paper.

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