Inverse modelling of thermal conductivity from temperature measurements at the Prototype Repository, Äspö HRL

it Article history: Received 25 June 2008 Accepted 26 January 2009 Available online 17 March 2009 Keywords: Thermal conductivity A¨spo¨ HRL Inverse modelling The Prototype Repository, at the A¨spo¨ HRL (Hard Rock Laboratory), is a demonstration project for the an outer (Fig. 1). A bentonite buffer is installed around the r the arger ed to erties of a as shown in Fig. 2. Laboratory and field measurements of thermal ARTICLE IN PRESS Contents lists availabl .els International & International Journal of Rock Mechanics & Mining Sciences 46 (2009) 1029–1041 ling. The inverse modelling uses an iterative process where � canisters in the deposition holes. The tunnel is backfilled with a mixture of bentonite and crushed rock. In 2001, the inner section was completed and the monitoring of processes started. The properties have been conducted within the prototype repository. A prognosis model of the thermal properties has been established based on these data. The prediction model is evaluated based on thermal property values back-calculated through inverse model- measured and calculated temperatures are fitted with a numerical model [2]. Corresponding author. Tel.: +4631187690; fax: +4613363091. E-mail address: [email protected] (J. Sundberg). 1365-16 doi:10.1 deposition sequence. The layout involves six deposition holes with electric heated canisters, four in an inner section and two in repository. Thirty-seven temperature sensors are located in the rock mass, for the deposition of spent nuclear fuel [1]. It provides a full-scale reference for testing predictive models relating to a repository, both for its individual components as well as for the complete system. It has also facilitated demonstration of the logistics of the conductivity. There is a maximum temperature limit fo buffer. If the rock has a low thermal conductivity, a l distance between the depositions holes is needed compar rock that is highly conductive. Thus, the thermal prop of the rock influence the size and construction cost 1. Introduction The Swedish Nuclear Fuel and Waste Management Company (SKB) is planning to build a final repository for all spent nuclear fuel. The Prototype Repository, situated about 500m below surface at the A¨spo¨ HRL (Hard Rock Laboratory), outside Oskarshamn, Sweden, is a reference and demonstration project installation of the outer section took place during 2003, and the surface between the outer plug and the rock was grouted in October 2004. The temperature distribution at the canisters, in the bentonite buffer and in the rock mass is influenced by the thermal properties of the rock mass, in particular its thermal 09/$ - see front matter & 2009 Elsevier Ltd. A 016/j.ijrmms.2009.01.012 relating to a spent nuclear fuel repository, both its individual components as well as the complete system. The final layout involves six deposition holes, four in an inner section and two in an outer, each fitted with an electrically heated canister. The access tunnel is backfilled with a mixture of bentonite and crushed rock. In 2001, the inner section was completed and monitoring of the heating process started. Temperature measurements in the rock mass are performed at 37 different points. In this paper, the measured thermal response in the surrounding rock is analysed by inverse modelling of the thermal conductivity of the rock mass. A three-dimensional finite difference model of the prototype repository (canisters, buffers, tunnel, etc.) is used to calculate the transient temperature increase due to the heat generation in the canisters. The value of a homogeneous rock thermal conductivity is chosen to obtain the best fit with measured data for each of the 37 temperature sensor points. The evaluation period for the fitting procedure is varied in order to study sensitivity to different time-scales. Measurements of thermal properties have been conducted within the prototype repository prior to the full-scale test. The thermal properties were predicted based on both field and laboratory measurements. These predictions are verified by comparison with thermal conductivity values calculated through inverse modelling. & 2009 Elsevier Ltd. All rights reserved. Received in revised form 22 December 2008 deposition of spent nuclear fuel, and provides a full-scale reference for testing predictive models Inverse modelling of thermal conductiv at the Prototype Repository, A¨spo¨ HRL Jan Sundberg a,�, Go¨ran Hellstro¨m b a Geo Innova AB, Arvid Hedvalls Backe 4, SE-411 33 Go¨teborg, Sweden b Division of Mathematical Physics, Lund University, SE-221 00 Lund, Sweden a r t i c l e i n f o a b s t r a c t journal homepage: www Rock Mechanics ll rights reserved. y from temperature measurements e at ScienceDirect evier.com/locate/ijrmms Journal of Mining Sciences ARTICLE IN PRESS ock J. Sundberg, G. Hellstro¨m / International Journal of R1030 2. Geology Lithological and fracture mapping for one part of the tunnel is shown in Fig. 3. The A¨spo¨ diorite is the dominant rock-type in the tunnel but subordinate rock types such as greenstone and fine- grained granite also occur. The A¨spo¨ diorite is dominated by the minerals plagioclase, alkali-feldspar, biotite and quartz. Fractures and fracture zones have been found along the tunnel. In the section studied, there is a zone of increased fracturing (45 fractures/m) at chainage 3520, while part of another zone appears at the end of the section (chainage 3600); see Fig. 3. Fig. 1. Schematic view of the layout of the Prototype Repository (not to scale) [1]. Fig. 2. Overview of the temperature sensors in the rock. Length section with the inner part of the prototype tunnel to the left (upper) and cross section towards the end of the tunnel (lower) [3]. the open tunnel. However, the results from the temperature measurements at the prototype repository are also influenced by water movements, especially in the initial stage after installation of the backfill, and should consequently influence the back- calculated thermal conductivities from the inverse modelling. The prediction of the effective thermal conductivity, it could be argued, should therefore be based on field measurements only. However, the water movements are probably highest close to the tunnel, so that field measurements made near the tunnel floor probably overestimate the effective thermal conductivity at a scale relevant for the thermal impact on the canister (cube with a side length of 5m). A combination of #3 (laboratory measure- ment-pure conduction) and #4 (field measurements—including a convective part) may be a reasonable prognosis (#5) of the effective thermal conductivity (2.62W/mK). However, laboratory measurements and field measurements represent different scales. Upscaling laboratory measurements to field measurement scale implies approximately the same mean but decreased variability. Nevertheless, this variance reduction has been neglected, due to sparse data, when combining the different types of measurements in #5. Further, the laboratory measurements of thermal conductivity were performed at room temperature (approximately 20–25 1C), whereas the field measurements were field measurements are made 0.6m below the tunnel floor and are possibly influenced by groundwater movements, 3. Prediction of thermal properties Prediction of the thermal properties has been made based on data from laboratory measurements on rock samples with the TPS method [5–7], and by direct field measurements [6] in the prototype tunnel. Measurements with the TPS method are performed by placing a thin sensor between two rock samples [5,7]. Field measurements are performed by the multi probe method using heating and temperature probes in parallel bore- holes of small diameters [7]. Different prognosis models were established depending on where and how the samples were obtained [2]: #1. Prototype repository section 1 (inner), laboratory measure- ments (six samples). #2. Prototype repository section 2 (outer), laboratory measure- ments (four samples). #3. Prototype repository sections 1 and 2, laboratory measure- ments (10 samples). #4. Prototype repository sections 1 and 2, field measurements (five samples). #5. Combination of prognosis 3 and 4 (15 samples). Probability plots to evaluate the distribution of data included in different models are presented in Fig. 4 for models 1–5. A summary of the prediction models for the thermal conductivity within the prototype repository is presented in Table 1. A good prediction of the thermal conductivity of the intact rock is judged to be #3, which is based on all laboratory measurements from the prototype repository (2.52W/mK). The number of measurements is quite small and there is no reason to believe that the variability detected in different parts of the Prototype Repository would not be representative of the inner part where the temperature measurements have been conducted (see Fig. 2). The results for the field measurements (2.83W/mK) are somewhat higher than the laboratory measurements. The partly induced by hydraulic gradient at 450m depth towards Mechanics & Mining Sciences 46 (2009) 1029–1041 performed at ambient temperature at 15 1C. However, the temperature dependence of thermal conductivity is small for the actual rock type [6]. ARTICLE IN PRESS ock J. Sundberg, G. Hellstro¨m / International Journal of R 4. Inverse numerical modelling 4.1. Measured data The material used in this study consists of the following measured data from thermal tests at the prototype repository: (1) canister power rates with an average sampling frequency for the six canisters varying from 17 to 24 readings per day. (2) Rock temperatures have been measured at 37 temperature sensor locations with roughly an hourly frequency, for 525 days. Temperature measurements from the start of canister heating were available for the evaluation. (3) Initial temperatures and coordinates for the 37 sensors. (4) Temperature on the inside and the outside of the canisters. 4.2. Location of temperature sensors In Fig. 5, the heat sources are indicated as blue rectangles, and the sensors are indicated as black and grey circles. 4.3. Conditions of buffer and backfill This study focuses on the thermal conditions in the rock mass. Thus, the conditions at the canister and in the bentonite buffer are not of primary interest in this study. Similar to the evaluation of thermal probe methods, it is assumed that these conditions have little influence on the evolution of rock temperatures, after a Fig. 3. Lithological and fracture map Mechanics & Mining Sciences 46 (2009) 1029–1041 1031 certain initial period. The heat transfer properties of the buffer and the capacities of the involved materials are of course important in the immediate vicinity of the canisters and during the initial heating period. Most of the heat released during this period is absorbed by the canister and buffer. A certain initial period is, therefore, ignored during the parameter fitting procedure. The water saturation of the bentonite buffers can be men- tioned as an example of the complexity of the boundary conditions close to canister. The bentonite buffers surrounding the canisters are fairly dry when the heating starts, but there is a rapid increase in water saturation as the experiment starts if water is available. Otherwise the increase in water saturation will be slow. Saturation of bentonite buffers around the canisters is predicted to take 2–3 years along the canister, and 5–6 years in the thicker bentonite below and above the canister. The relative humidity of the sensors indicates that the bentonite is close to water saturation after 1 year of operation although the conditions seem to be fairly heterogeneous [3]. The tunnel above the deposition holes is backfilled with a mixture of bentonite (30%) and crushed rock. Since the tunnel was backfilled it has gradually become water-saturated. 4.4. Evaluation method The measured thermal response in the surrounding rock is analysed by inverse modelling of the thermal properties of the ping of the prototype tunnel [4]. ARTICLE IN PRESS l - 9 or p ock 99 90 99 90 Model 1 Norma Probability plots f J. Sundberg, G. Hellstro¨m / International Journal of R1032 rock mass. In this study, a three-dimensional finite difference model of the prototype repository (canisters, buffers, tunnel, etc.) is used to calculate the transient temperature increase due to the heat generation in the canisters. The value of a homogeneous rock thermal conductivity is varied until the best fit with measured data is obtained for each sensor point. The evaluation period for the fitting procedure is varied in order to study sensitivity to different time-scales. The heat capacity is given and kept constant. Ideally the heat capacity should be evaluated from the fitting procedure. However, the sensitivity is small in this pseudo steady- state process. Pe rc en t 2.82.62.42.2 50 10 1 2.0 50 10 1 3.002.752.502.252.00 99 90 50 10 1 2.2.0 99 90 50 10 1 3.53.02.52.0 99 90 50 10 1 Model 3 Model 5 Fig. 4. Probability distribution (lognormal distributed) of five thermal conductivity Table 1 Summary of results from prediction modelling of thermal properties of the A¨spo¨ diorit Prognosis Population #1 Prototype repository section 1 (inner), laboratory measurements #2 Prototype repository section 2 (outer), laboratory measurements #3 Prototype repository section 1+2, laboratory measurements #4 Prototype repository section 1+2, field measurements #5 Prototype repository section 1+2, field and laboratory measuremen Combination of prognosis #3 and #4 Model 2 Model 1 Mean 2.438 StDev 0.07834 N AD 5% CI rognosis models 6 0.145 Mechanics & Mining Sciences 46 (2009) 1029–1041 4.5. Numerical model The thermal process in a repository is governed by the heat conduction equation: l q2T qx2 þ q 2T qy2 þ q 2T qz2 ! þ q ¼ C qT qt (1) where l is the thermal conductivity (W/mK), C is the volumetric heat capacity (J/m3K), and q is a heat source (W/m3). Numerical values of the thermal properties are given in Table 1. 3.22.82.4 3.63.22.84 Model 4 P-Value0.929 Model 2 Mean StDev N AD 0.313 P-Value Model 3 Mean StDev N AD P-Value Model 4 Mean StDev N 5 AD P-Value Model 5 Mean StDev N AD P-Value 2.633 0.1517 4 0.333 2.516 0.1454 10 0.376 0.338 2.828 0.1923 0.582 0.060 2.62 0.2176 15 0.415 0.292 (W/mK) datasets (models 1–5) of A¨spo¨ diorite from the prototype repository. e [2]. Mean W/(mK) Std. dev. W/(mK) Number of samples Distribution 2.44 0.08 6 Lognormal 2.63 0.15 4 Lognormal 2.52 0.15 10 Lognormal 2.83 0.19 5 Lognormal ts. 2.62 0.22 15 Lognormal assumed to be conductive only. Thus, convection caused by the In the second step, the simulated values are interpolated in time to match the times at which the measured data were collected for each sensor. The third step involves comparison of the measured and simulated temperatures for each point and for each thermal conductivity during the chosen evaluation interval. To do this, the average temperature of the measured temperature and the average of the simulated superimposed temperature increase during the evaluation period is first calculated. The difference between these two averages is assumed to be the initial temperature component of the simulated temperature field. The ARTICLE IN PRESS PXPTA0320 PXPTA0620 PTA3 XPTA XPTA XPTA PTA PXPTA1820 PXPTA2120 2310 2150 40 tunn parameter. ock Mech inflow to the backfilled tunnel is ignored. All thermal properties are constant, which means that the thermal problem is mathematically linear. Different solutions to the heat conduction problem can then be superimposed on each other to form the complete temperature field. Here, this technique is used by superimposing two parts of the thermal response—the initial temperature field and the temperature increase due to the heat generation in the canisters. The initial temperature is The measured time-varying canister power rate is injected uniformly by the heat source term q into each canister. The numerical model of the repository uses an explicit finite difference scheme [8]. The simulated ground region encompasses a parallelepipedical volume of 120�150�120m3 and is de- scribed with a grid using 54�197�59 ¼ 627642 cells for the numerical simulation scheme. 4.6. Assumptions The thermal properties of the different materials involved in the large-scale thermal process are assumed to be homogeneous. The thermal properties are given in Table 2. The heat transport is PXPTA0310 PXPTA0610PXPTA1010 PXPTA1020 PXPTA1810 PX P PXPTA2410 PXPTA2110 PXPTA2420 Fig. 5. Location of temperature sensors [2]. The inner part of the prototype PXPTA0330 PXPTA0340 PXPTA0350 PXPTA0630 PXPTA0640 PXPTA0650 PXPTA1030 PXPTA1040 PXPTA1050 P P PXPXPTA1840 PXPTA1830 PXPTA1850 PXPTA2140 PXPTA2130 PXPTA2430 PXPTA2320 PXPTA PXPTA2330 PXPTA PXPTA24 Tunnel J. Sundberg, G. Hellstro¨m / International Journal of R assumed to be uniform, although it is apparent from the initial rock temperatures that there is also a certain superimposed thermal disturbance from activities in the deposition tunnel. Since this disturbance will decline with time, the influence on the evaluation will be reduced if a certain initial time period is omitted from the fitting procedure. The initial temperature in the calculation of the superimposed temperature increase due to canister heating is set to zero everywhere. 4.7. Fitting procedure The measured thermal response is used to find the thermal conductivity that results in the best fit with the simulated thermal response. The thermal conductivity of the rock is set to a constant homogeneous value for each calculation of the temperature disturbance resulting from heat generation within each canister. The value of thermal conductivity is varied from 1.9 to 3.7W/mK with an increment of 0.1W/mK. During the simulation for each value of thermal conductivity, the temperature increase at each temperature sensor location is calculated. The temperatures at 12-hour time intervals for a total simulation time of 730 days are recorded and stored. Legend Heat source Temperature sensors (not aligned with cross section) Temeprature sensors 010 3020 3030 3040 el is to the left. The numbering of canisters is from the left. See also Fig. 2. Table 2 Thermal properties of the materials involved in the thermal process. Thermal conductivity (W/mK) Volumetric heat capacity (MJ/m3K) Canister 15.0 4.00 Buffer 1.5 3.40 Tunnel backfill 1.5 2.50 Rock To be estimated 2.2a [6] a A value of 2.31 (MJ/m3K) was also used in order to test the sensitivity to this 3050 sim tem incr sim per ther valu for betw foun T con (a) anics & Mining Sciences 46 (2009) 1029–1041 1033 ulated temperature field is then the sum of this initial perature component and the superimposed temperature ease. The square of the difference between measured and ulated temperature for each measured time in the evaluation iod is summed. This procedure is repeated for each value of mal conductivity. Finally, we have the square sum for 19 es of thermal conductivity in the range from 1.9 to 3.7W/mK each sensor. The thermal conductivity of the best match een measured and simulated values for a given sensor is d by minimizing a polynomial fit to the 19 values. he choice of evaluation period is guided by three different cerns: To minimize the influence of an incomplete description of the local conditions in the canister and bentonite buffer, a certain initial period should not be included in the fitting procedure. For instance, the varying saturation level in the bentonite buffer will influence the local heat transfer characteristics. Lower heat transfer capacity gives higher canister tempera- ture and thus more energy is stored in the canister and less is released to the surroundings. It is apparent from the measured canister temperature that the temperature rise is most pronounced in the beginning. Later on, the temperature T stud reas exp tem two sim cho a ce is o star case per F ave betw sim to a inte som The deviations in the initial temperatures will be less for sensors close to the canisters (Fig. 9). The difference between the measured temperature and the simulated temperature increase may be regarded as the ‘‘fitted’’ initial temperature of the simulation. The fitted initial simulation temperature is compared with the measured initial temperature in Fig. 10. The measured initial temperatures range between 14 and 18 1C, with a tendency towards higher values close to the tunnel and the deposition boreholes. The original undisturbed rock temperature is likely to have been much more uniform in this relatively limited region. With a vertical gradient of about 15 1C/km, the range of initial temperature at the probes would have been within 0.2 1C. It is evident from the measurements that there are ‘‘residual’’ temperature disturbances from the local climate in the deposition tunnel. These disturbances, which can be treated as a superimposed thermal process, will decline with time. The examination of the fitting procedure for thermal probe PXPTA2120 (see Figs. 6 and 7) indicates that omitting a longer initial time period from the fitting is likely to result in larger difference between fitted and measured initial temperature. Shifting the evaluation from an early period to a later one tends ARTICLE IN PRESS PXPTA2120 Evaluation period 160-525 days 15 20 25 30 35 40 45 50 Days Te m pe ra tu re Measured Simulated 5004003002001000 Fig. 7. Thermal response (1C) of thermal probe PXPTA2120 over a period of 525 days. The period 160–525 days is used for fitting the measured and simulated response. ock ed in decreasing order according to magnitude of average ulated temperature increase for each sensor. Points belonging certain case or category are connected by a line for easier 525 sort luation period at 160 days results in a better fit during the sen evaluation period than setting the time to 30 days. There is rtain discrepancy during the initial period before 160 days. It f course more difficult to obtain a better fit for the period ting at 30 days since the period is considerably longer. In this , a noticeable deviation occurs at the end of the evaluation iod. ig. 8 shows the average measured rock temperature, the rage simulated temperature increase and the difference een these values during the evaluation period from 160 to days for all 37 temperature sensors. These results have been A eva he influence of the duration of the evaluation period is ied for two different start values: 30 days and 160 days. The ons for omitting a certain initial time period have been lained above. The point PXPTA2120, which shows the largest perature change, is used as a reference. A comparison for the cases is made with an end time of 525 days. The fitted ulated thermal responses are given in Figs. 6 and 7. general observation is that setting the start time for the becomes more stable and the energy absorbed in the canister decreases. (b) Local disturbances of the initial temperatures. These distur- bances are due to activities and varying temperatures in both the deposition holes and the tunnel during construction. The distances from the surfaces to which these temperature changes penetrate depend on the duration of exposure. The variation in measured initial temperature is an indication of this process. After the sealing of the tunnel there will be a decline in the disturbances on a time-scale similar to the exposure duration. The chosen fitting procedure may result in simulated ‘‘initial’’ temperatures being slightly different than the measured ones. (c) Another important process may be convective heat transport due to large-scale groundwater flow in the fissures. The principal influence is most pronounced on a long time scale. Using a fitting procedure based on conductive heat transport will lead to increasingly higher ‘‘effective’’ thermal conductiv- ities when longer evaluation periods are considered in order to compensate for the energy transport away from the rock around the canisters. It should be emphasized that the fitted thermal conductivity values are calculated for each sensor point individually without regard to the thermal response of any other point. However, it is also possible to achieve an overall thermal conductivity by considering the best fit for all 37 sensor points. The procedure to obtain this overall thermal conductivity value is carried out as follows. The average square sum of the difference between measured and calculated temperatures for each of the 37 sensor points are summed up for each value of the thermal conductivity (1.9 to 3.7W/mK). The thermal conductivity of the best overall match between measured and simulated values taking all sensors into account is found by minimizing a polynomial fit to the 19 values. 4.8. Results of evaluated thermal conductivity 4.8.1. Influence of duration of evaluation period and initial temperatures J. Sundberg, G. Hellstro¨m / International Journal of R1034 rpretation of trends. The temperature increase reflects to e degree the proximity of a temperature sensor to a canister. influence of a relative error in the measurements and local PXPTA2120 Evaluation period 30-525 days 15 20 25 30 35 40 45 50 5004003002001000 Days Te m pe ra tu re Measured Simulated Fig. 6. Thermal response (1C) of thermal probe PXPTA2120 over a period of 525 days. The period 30–525 days is used for fitting the measured and simulated response. Mechanics & Mining Sciences 46 (2009) 1029–1041 to give higher fitted thermal conductivities, which may be the result of a general temperature decrease or convective heat transport due to groundwater flow. The largest difference between ARTICLE IN PRESS ock 45 J. Sundberg, G. Hellstro¨m / International Journal of R measured and fitted initial temperatures is found for probes PXPTA2140 and PXPTA1830, which are situated fairly close to each other on the same vertical level (see Fig. 5). Also the neighbouring probes PXPTA2130 and PXPTA1850 show a large difference. The average deviation between measured and simu- 0 5 10 15 20 25 30 35 40 PX PT A2 12 0 PX PT A3 03 0 PX PT A1 82 0 PX PT A2 11 0 PX PT A2 13 0 PX PT A3 02 0 PX PT A2 43 0 PX PT A1 03 0 PX PT A2 14 0 PX PT A2 42 0 PX PT A1 83 0 PX PT A1 02 0 PX PT A2 32 0 PX PT A3 01 0 PX PT A1 04 0 PX PT A1 84 0 PX PT A1 85 PX PT PX PT A3 04 0 PX PT A1 81 0 Te m pe ra tu re Fig. 8. Average measured rock temperature, average simulated temperature (1C) incre temperature sensors. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 A ve ra ge d ev ia tio n/ te m pe ra tu re c ha ng e PX PT A2 12 0 PX PT A3 03 0 PX PT A1 82 0 PX PT A2 11 0 PX PT A2 13 0 PX PT A3 02 0 PX PT A2 43 0 PX PT A1 03 0 PX PT A2 14 0 PX PT A2 42 0 PX PT A1 83 0 PX PT A1 02 0 PX PT A2 32 0 PX PT A3 01 0 PX PT A1 04 0 PX PT A1 84 0 PX PT A1 PXPX PT A3 04 0 PX PT A1 81 0 Fig. 9. Average deviation between measured and simulated temperatures (1C) in relat period 160–365 days is used for fitting the measured and simulated response. Mechanics & Mining Sciences 46 (2009) 1029–1041 1035 lated temperatures, in relation to simulated temperature increase for each of the 37 temperature sensors, lie in the range 0.2–1.2% for those points where the temperature increase exceeds 5 1C. For the five points with the smallest increase, less than 2 1C, the deviation is 2–6%. 0 A3 05 0 PX PT A2 33 0 PX PT A2 44 0 PX PT A2 15 0 PX PT A1 05 0 PX PT A0 63 0 PX PT A0 62 0 PX PT A2 41 0 PX PT A2 31 0 PX PT A0 61 0 PX PT A0 65 0 PX PT A0 33 0 PX PT A0 32 0 PX PT A0 34 0 PX PT A0 35 0 PX PT A1 01 0 PX PT A0 64 0 PX PT A0 31 0 Measured average temperature Average simulated temperature change Temperature difference ase and difference during the evaluation period 160–525 days for each of the 37 85 0 PT A3 05 0 PX PT A2 33 0 PX PT A2 44 0 PX PT A2 15 0 PX PT A1 05 0 PX PT A0 63 0 PX PT A0 62 0 PX PT A2 41 0 PX PT A2 31 0 PX PT A0 61 0 PX PT A0 65 0 PX PT A0 33 0 PX PT A0 32 0 PX PT A0 34 0 PX PT A0 35 0 PX PT A1 01 0 PX PT A0 64 0 PX PT A0 31 0 ion to simulated temperature increase for each of the 37 simulated sensors. The ARTICLE IN PRESS Measured initial temperature 84 0 PT A1 8 PX P ) fo ock 4.8.2. Summary of results for different evaluation periods The rock thermal conductivity that gives the best fit between measured and simulated temperatures for each of the 37 temperature sensors is presented in Fig. 11 for the three chosen evaluation periods 160–250, 160–365, and 160–525 days. A definite trend towards higher values of the fitted thermal conductivity can be noted for almost all points when the end time of the evaluation period is extended. The influence of the start time for the evaluation is compared 10 11 12 13 14 15 16 17 18 19 20 Te m pe ra tu re PX PT A2 12 0 PX PT A3 03 0 PX PT A1 82 0 PX PT A2 11 0 PX PT A2 13 0 PX PT A3 02 0 PX PT A2 43 0 PX PT A1 03 0 PX PT A2 14 0 PX PT A2 42 0 PX PT A1 83 0 PX PT A1 02 0 PX PT A2 32 0 PX PT A3 01 0 PX PT A1 04 0 PX PT A1 PXPX PT A3 04 0 PX PT A1 81 0 Fig. 10. Measured initial temperature and fitted ‘‘initial’’ temperature (1C J. Sundberg, G. Hellstro¨m / International Journal of R1036 for a fixed end time of 525 days in Fig. 12. Changing the start time from 30 to 160 days yields higher values of the fitted thermal conductivities for almost all points. 4.8.3. Influence of volumetric heat capacity The influence of the volumetric heat capacity of the rock has been assessed by performing a calculation with a 5% higher value than the reference case. As can be expected, the slope of the response curve is essentially determined by the thermal con- ductivity, whereas the fitted ‘‘initial’’ temperature has to be raised to compensate for the increased capacitive effect of the rock. 4.8.4. Influence of large-scale temperature change Previous and ongoing activities in the tunnels (e.g. ventilation) may have disturbed the rock temperatures during the monitoring period. If the air temperature in the tunnel during excavation and instrumentation was higher than the initial undisturbed rock temperatures, it would have caused a temperature increase in the rock close to the tunnel. The thermally disturbed zone would have grown slowly with time until the tunnel was backfilled. It is documented that the air temperature near the prototype repository fluctuates during the year with an amplitude of about 10 1C (max/min temp approximately 20/10 1C). The altered geohydrological conditions around the tunnel will also cause groundwater movements and associated convective heat transfer on a large scale. To check the possible influence of a slow large-scale (global) temperature drift, a general constant temperature change of �0.2 1C/year has been superimposed. The simulated temperature increase (slope) will then become lower for a given thermal conductivity. In order to compensate for this effect, the fitting procedure will find a lower thermal conductivity (Fig. 13). The influence is larger for temperature sensors farther away from the canisters where the temperature increase due to canister heating is small. Conversely, a constant global temperature increase will result in higher thermal conductivities. Compared with the case without global temperature change, there is a trend towards lower values of the fitted thermal 50 TA 30 50 PX PT A2 33 0 PX PT A2 44 0 PX PT A2 15 0 PX PT A1 05 0 PX PT A0 63 0 PX PT A0 62 0 PX PT A2 41 0 PX PT A2 31 0 PX PT A0 61 0 PX PT A0 65 0 PX PT A0 33 0 PX PT A0 32 0 PX PT A0 34 0 PX PT A0 35 0 PX PT A1 01 0 PX PT A0 64 0 PX PT A0 31 0 r each of the 37 temperature sensors. The period 160–525 days is used. Simulated initial temperature Mechanics & Mining Sciences 46 (2009) 1029–1041 conductivity especially for distant sensors with small temperature increase during the monitoring period. The relative influence of the global temperature change is larger for these sensors. The fitted thermal conductivity values still become larger as the duration of the evaluation period increases. 4.8.5. Results in relation to location of temperature sensors In Fig. 14, the thermal conductivity achieved from curve fitting using data from 160 through 525 days after the initiation of the experiment were used. Fig. 15 shows the situation when a slow large-scale temperature drift is included. The identities of individual temperature sensors are shown in Fig. 5. High thermal conductivity values are modelled in the inner parts of the tunnel, especially close to the tunnel surface. These high values may be caused by water movements, which have been reported in the actual parts of the tunnel. Due to the limited period of heating, evaluated thermal conductivity values close to the canister are probably the most reliable. However, the computed initial temperatures may be an indication of water movements. Differences between measured and fitted initial temperatures larger than 2 1C are shown in Fig. 16. The largest differences (43 1C) between measured and fitted initial temperatures are found for probes PXPTA2140, PXPTA1830 and PXPTA1040. The first two are fairly close on the same vertical level. The computed initial temperatures are higher than the measured ones for all probes except for probes PXPTA2410 and PXPTA2420. A component of convective heat transport is a probable explanation for these differences. ARTICLE IN PRESS ock 3.4 3.5 3.6 3.7 3.8 Evaluation period 160-250 days Evaluation period 160-365 days Evaluation period 160-525 days J. Sundberg, G. Hellstro¨m / International Journal of R 5. Verification of predictive model The measured thermal response has been used to estimate the effective thermal conductivity by finding the best fit with results obtained by a numerical simulation of the thermal process around the repository. Table 3 presents the thermal conductivity that gives the best fit considering all temperature sensors. The sensitivity of the fitted 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 Th er m al c on du ct iv ity (W /(m ,K )) PX PT A2 12 0 PX PT A3 03 0 PX PT A1 82 0 PX PT A2 11 0 PX PT A2 13 0 PX PT A3 02 0 PX PT A2 43 0 PX PT A1 03 0 PX PT A2 14 0 PX PT A2 42 0 PX PT A1 83 0 PX PT A1 02 0 PX PT A2 32 0 PX PT A3 01 0 PX PT A1 04 0 PX PT A1 84 0 PX PT A1 8 PXPX PT A3 04 0 PX PT A1 81 0 Fig. 11. Rock thermal conductivity that gives the best fit between measured and simula 160–250 days, 160–365 days, or 160–525 days is used. 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Th er m al c on du ct iv ity (W /(m ,K )) Evaluation period 30-525 days Evaluation period 160-525 days PX PT A2 12 0 PX PT A3 03 0 PX PT A2 13 0 PX PT A2 11 0 PX PT A1 82 0 PX PT A3 02 0 PX PT A1 03 0 PX PT A3 04 0 PX PT A2 14 0 PX PT A1 02 0 PX PT A1 83 0 PX PT A2 42 0 PX PT A2 32 0 PX PT A1 04 0 PX PT A3 01 0 PX PT A1 84 0 PX PT A1 8 PXPX PT A2 43 0 PX PT A1 81 0 Fig. 12. Rock thermal conductivity that gives the best fit between measured and simul 30–525 days or 160–525 days is used. Mechanics & Mining Sciences 46 (2009) 1029–1041 1037 thermal conductivity with regards to the volumetric heat capacity is small. The overall influence of a large-scale (global) temperature drift is also fairly small, although it becomes relatively important for temperature sensors showing only a little increase in temperature due to canister heating. In Table 4 comparisons are made between different prognoses (Table 1) and inverse modelling results for all sensors. We have argued above that a reasonable prediction of the effective thermal 50 PT A3 05 0 PX PT A2 33 0 PX PT A2 44 0 PX PT A2 15 0 PX PT A1 05 0 PX PT A0 63 0 PX PT A0 62 0 PX PT A2 41 0 PX PT A2 31 0 PX PT A0 61 0 PX PT A0 65 0 PX PT A0 33 0 PX PT A0 32 0 PX PT A0 34 0 PX PT A0 35 0 PX PT A1 01 0 PX PT A0 64 0 PX PT A0 31 0 ted temperatures for each of the 37 temperature sensors. An evaluation period of 50 PT A3 05 0 PX PT A2 33 0 PX PT A1 01 0 PX PT A2 44 0 PX PT A1 05 0 PX PT A0 63 0 PX PT A0 62 0 PX PT A2 41 0 PX PT A2 31 0 PX PT A0 61 0 PX PT A0 65 0 PX PT A0 33 0 PX PT A0 32 0 PX PT A0 34 0 PX PT A0 35 0 PX PT A2 15 0 PX PT A0 64 0 PX PT A0 31 0 ated temperatures for each of the 37 temperature sensors. An evaluation period of ARTICLE IN PRESS ock 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 J. Sundberg, G. Hellstro¨m / International Journal of R1038 conductivity is a combination of laboratory (thermal conductivity) and field measurements (effective thermal conductivity) at the prototype repository (#5). In Table 5, comparison is also made of the prediction with the best fit of inverse modelling to individual temperature sensors, including standard deviation for both types of distributions. The standard deviation is calculated from the thermal conductivity values based on best fit between measured and simulated temperature for individual temperature sensors. Prognosis #5 (Table 2) displays good agreement with back- calculated thermal conductivities from the inverse modelling and the above statement of a reasonable prediction of the effective thermal conductivity seems to be verified. 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Th er m al c on du ct iv ity (W /(m ,K )) Global temperature change 0.0 C/year Global temperature change -0.2 C/year PX PT A2 12 0 PX PT A3 03 0 PX PT A1 82 0 PX PT A2 11 0 PX PT A2 13 0 PX PT A3 02 0 PX PT A2 43 0 PX PT A1 03 0 PX PT A2 14 0 PX PT A2 42 0 PX PT A1 83 0 PX PT A1 02 0 PX PT A2 32 0 PX PT A3 01 0 PX PT A1 04 0 PX PT A1 84 0 PX PT A1 8 PXPX PT A3 04 0 PX PT A1 81 0 Fig. 13. Rock thermal conductivity that gives the best fit between measured and simulat 160–525 days. The global temperature change is 0 or �0.2 1C/year. PXPTA0310 PXPTA0320 PXPTA0330 PXPTA0340 PXPTA0350 PXPTA0610 PXPTA0620 PXPTA0630 PXPTA0640 PXPTA0650 PXPTA1010 PXPTA1020 PXPTA1030 PXPTA1040 PXPTA1050 PXPTA1810 PXPT PXPT PXPTA PXPT PXPT PXPTA1840 PXPTA1830 PXPTA1820 PXPTA2410 PXPTA1850 PXPTA2140 PXPTA2130 PXPTA2120 PXPTA2110 PXPTA2420 PXPTA2430 PXPTA2320 PXPTA2310 PXPTA2330 PXPTA2150 PXPTA2440 Tunnel Fig. 14. Thermal conductivity from simulation inc Mechanics & Mining Sciences 46 (2009) 1029–1041 6. Discussion The results from the inverse modelling indicate large-scale temperature disturbances in the rock mass. The uneven tempera- ture distribution in the studied region at the start of experiment is a remnant of the influence from tunnels and boreholes during the preceding construction phase. This statement is strengthened by the fact that measurements from 1999 (field measurements of thermal conductivity) show a temperature of about 11.5–12.5 1C (at 0.6m below rock surface), which can be compared with an initial temperature of 14–17 1C for the present experiment. This indicates that an overall decrease of the undisturbed temperature field during the experiment is a realistic assumption. Rather large 50 PT A3 05 0 PX PT A2 33 0 PX PT A2 44 0 PX PT A2 15 0 PX PT A1 05 0 PX PT A0 63 0 PX PT A0 62 0 PX PT A2 41 0 PX PT A2 31 0 PX PT A0 61 0 PX PT A0 65 0 PX PT A0 33 0 PX PT A0 32 0 PX PT A0 34 0 PX PT A0 35 0 PX PT A1 01 0 PX PT A0 64 0 PX PT A0 31 0 ed temperatures for each of the 37 temperature sensors for an evaluation period of Legend Heat source Thermal Conductivity (sensors not aligned with the cross section are marked with grey font) A3010 A3020 3030 A3040 A3050 3.5 W/(m K) 3 W/(m K) 2.5 W/(m K) luding curve fitting for 160 through 525 days. ARTICLE IN PRESS ock J. Sundberg, G. Hellstro¨m / International Journal of R annual oscillations of air temperature have also been measured near the prototype repository with amplitudes of about 10 1C. The accuracy of the evaluation would have been improved if the rock PXPTA0310 PXPTA0320 PXPTA0330 PXPTA0340 PXPTA0350 PXPTA0610 PXPTA0620 PXPTA0630 PXPTA0640 PXPTA0650 PXPTA1010 PXPTA1020 PXPTA1030 PXPTA1040 PXPTA1050 PXPTA1810 PXPTA PXPTA PXPTA PXPTA PXPTA PXPTA1840 PXPTA1830 PXPTA1820 PXPTA2410 PXPTA1850 PXPTA2140 PXPTA2130 PXPTA2120 PXPTA2110 PXPTA2420 PXPTA2430 PXPTA2320 PXPTA2310 PXPTA2330 PXPTA2150 PXPTA2440 Tunnel Fig. 15. Thermal conductivity from simulation including curve fitting for 160 through 5 Fig. 16. Differences between measured and fitted initial temperatures larger than 2 1C in legend, the reader is referred to the web version of this article.) Table 3 Evaluated thermal conductivity from inverse modelling that gives the best fit considering all temperature sensors. Case Evaluation period (days) Thermal conductivity (W/mK) Reference 30�250 2.46 Reference 30�365 2.50 Reference 30�525 2.57 Reference 160�250 2.51 Reference 160�365 2.58 Reference 160�525 2.72 Volumetric heat capacity +5% 160�525 2.73 Global temp. change �0.2 C/year 160�250 2.47 Global temp. change �0.2 C/year 160�365 2.53 Global temp. change �0.2 C/year 160�525 2.65 Legend Heat source Thermal Conductivity 3010 3020 3030 3040 3050 3 W/(m K) 3.5 W/(m K) Mechanics & Mining Sciences 46 (2009) 1029–1041 1039 temperature had been allowed to equilibrate after the sealing of the tunnel and verified to be stable before the heating of the canisters began. It may also have made it possible to determine if there was a long-term large-scale temperature drift. The influence of the uncertainty of the initial tempe- rature on the estimated effective thermal conductivity can be reduced in the fitting procedure by excluding data from an initial period. Extending the duration of the measurement would allow more initial data to be omitted in order to improve accuracy and would also enhance prediction of groundwater flow effects. Prognosis #3 relies on precise laboratory measurements of thermal conductivity of rock samples from the prototype repository. However, these measurements do not account for a possible enhancement of heat transport due to groundwater movement that may occur in the field. A small contribution of convective heat transfer can be modelled as an increased effective thermal conductivity value. The results for the field measure- ments from the prototype repository (prognosis #4) are indeed somewhat higher than the laboratory measurements. The influ- ence of groundwater movement is probably more pronounced 2.5 W/(m K) (sensors not aligned with the cross section are marked with grey font) 25 days including a correction for global temperature change of �0.2 1C annually. dicated with green. (For the interpretation of the reference to colour in this figure ARTICLE IN PRESS K) Inverse modelling (160–525 days) 2.72 Inverse modelling (160–525 days), including possible 2.65 fers ear ear, osis ock during the early stage when the initially dry backfill becomes saturated, and close to the tunnel floor where the field measure- ments were conducted. Therefore, prognosis #5 is judged to be a reasonable estimate (a weighted mean value of the two prognoses #3 and #4). global temp. change �0.2 1C/year Laboratory measurements section 1+2 (prognosis #3) Combination of prognosis #3 and #4 (prognosis #5) Best fit to all sensors. Most relevant values are in bold (judgement). ‘‘Prognosis’’ re Table 5 Modelled thermal conductivity compared with prognosis (mean values). Case Inverse modelling (160–525 days) Inverse modelling (160–525 days), values above 3.4 excluded Inverse modelling (160–525 days), including possible global temp. change –0.2 1C/y Inverse modelling (160–525 days), including possible global temp. change �0.2 1C/y values above 3.4 excluded Laboratory measurements section 1+2 (prognosis #3) Combination of prognosis #3 and #4 (prognosis #5) Best fit to individual sensors. Most relevant values are in bold (judgement). ‘‘Progn Table 4 Modelled thermal conductivity compared with prognosis (mean values). Case Modelled thermal conductivity (W/m J. Sundberg, G. Hellstro¨m / International Journal of R1040 The verification of the prediction of thermal conductivity is judged to be good as regards best fit for all temperature sensors. However, for a few sensors there is a large discrepancy between the individual and the overall best fit. The five sensors with the largest square-sum deviations are located at the inner end of the tunnel. They exhibit the smallest temperature increase and are therefore most sensitive to disturbances. Water-bearing fractures are present in this region and local field measurements of thermal conductivity are obviously influenced by groundwater move- ments. The evaluated effective thermal conductivities exceed an improbable value of 3.5 for sensors near the floor in the inner part of the tunnel. These high values are most certainly a result of increased sensibility to disturbances due to low temperature rise and disturbances caused by groundwater movements, which are probably more pronounced close to the tunnel surface. If these improbably high values (above 3.4W/mK) are excluded from the mean, the result is close to the prediction. The simulations show that a there is a slight increase in effective thermal conductivity values if the evaluation period is prolonged. This may be caused by either groundwater movement or a large-scale temperature drift. A more realistic and stable fit is achieved if a constant negative temperature drift is introduced. 7. Conclusions There is good agreement between the prediction of thermal conductivity in the prototype repository and the result from inverse modelling based on 37 different temperature sensors. The rather low thermal conductivity value of around 2.6W/mK is verified. Some of the temperature measurements seem to be influenced by groundwater movements. The estimated thermal conductivity gives an effective value that includes a small contribution from convection and is, therefore, probably over- estimating the actual thermal conductivity. The variability in thermal conductivities from best fit of individual sensors seems to be the same as the variability in the prognoses that are based on Prognosis (W/mK) Comment 2.52 ‘‘Best’’ prediction of thermal conductivity 2.62 ‘‘Best’’ prediction of effective thermal conductivity to Table 1. Modelled thermal conductivity (W/mK) Std. (W/mK) Prognosis (W/mK) Std. (W/mK) 2.91 (N ¼ 37) 0.34 2.81 (N ¼ 33) 0.22 2.73 (N ¼ 37) 0.32 2.65 (N ¼ 34) 0.18 2.52 0.15 2.62 0.22 ’’ refers to Table 2. Mechanics & Mining Sciences 46 (2009) 1029–1041 direct measurements. The results provide no conclusive support for reduction of spatial variability as a result of upscaling. However, the analysis is simplified and more work is needed to determine the scale dependence with more confidence. There is a large discrepancy between the individual fit for some sensors and the overall best fit. The heat transport in the vicinity of these sensors is likely to be influenced by groundwater movements and thus the thermal conductivity is overestimated. The relatively wide range of measured initial temperature shows that there is a considerable thermal disturbance, which can be explained by the varying air temperatures in the tunnels and in the boreholes during the construction phase. The accuracy of the evaluation would have been improved if the rock temperature had been allowed to equilibrate after the sealing of the tunnel and verified to be stable before the heating of the canisters began. The results of the inverse modelling at the prototype repository indicate that data is influenced by a temperature drift in the rock mass and/or by water movements. These ‘‘errors’’ are probably most significant in early data. It is recommended that data from a longer period of time are evaluated. Extending the duration of the measurement would also allow more initial data to be omitted in order to improve accuracy. This would also enhance prediction of groundwater flow effects. Furthermore, the evaluation of up-scaling effects on the thermal conductivity distribution from the inverse modelling would be improved if a longer period is evaluated. It is possible to make predictions of the thermal conductivity from laboratory measurements. For a full scale repository this implies such prognoses can be used in order to design the distance between deposition holes. If the thermal conductivity is at a critical level (low conductivity) or the variability is high, local measurements close to each deposition hole are needed to confirm the prognosis. References [1] SKB. A¨spo¨ Hard Rock Laboratory. Annual report 2004. Report TR-05-10. Stockholm: Svensk Ka¨rnbra¨nslehantering AB; 2005. [2] Sundberg J, Back PE, Hellstro¨m G. Scale dependence and estimation of rock thermal conductivity. Analysis of upscaling, inverse thermal modelling and value of information with the A¨spo¨ HRL prototype repository as an example. Report SKB R-05-82. Stockholm: Svensk Ka¨rnbra¨nslehantering AB; 2005. [3] Goudarzi R, Johannesson L. A¨spo¨ HRL, Prototype Repository. Sensors data report (period 010917-040301). Report no. 9, SKB IPR-04-24. Stockholm: Svensk Ka¨rnbra¨nslehantering AB; 2004. [4] Rhe´n I, editor. Documentation of tunnel and shaft data. Tunnel section 2874–3600m, hoist and ventilation shafts 0–450m. Progress report 25-95-28. Stockholm: Svensk Ka¨rnbra¨nslehantering AB; 1995. [5] Gustafsson S. Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solid materials. Rev Sci Instrum 1991;62: 797–804. [6] Sundberg J, Gabrielsson A. Laboratory and field measurements of thermal properties of the rocks in the prototype repository at A¨spo¨ HRL. Report SKB IPR-99-17. Stockholm: Svensk Ka¨rnbra¨nslehantering AB; 1999. [7] Sundberg J. Determination of thermal properties of A¨spo¨ HRL. Compari- son and evaluation of methods and methodologies for borehole KA2599G01. Report SKB R-02-27. Stockholm: Svensk Ka¨rnbra¨nslehantering AB; 2002. [8] Eftring B. Numerisk bera¨kning av temperaturfo¨rlopp. Ett fysikaliskt betraktelsesa¨tt. (Numerical calculation of thermal processes. A physical approach). PhD thesis, Department of Building Science, Lund University; 1990. ARTICLE IN PRESS J. Sundberg, G. Hellstro¨m / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 1029–1041 1041 Inverse modelling of thermal conductivity from temperature measurements at the Prototype Repository, Äspö HRL Introduction Geology Prediction of thermal properties Inverse numerical modelling Measured data Location of temperature sensors Conditions of buffer and backfill Evaluation method Numerical model Assumptions Fitting procedure Results of evaluated thermal conductivity Influence of duration of evaluation period and initial temperatures Summary of results for different evaluation periods Influence of volumetric heat capacity Influence of large-scale temperature change Results in relation to location of temperature sensors Verification of predictive model Discussion Conclusions References