1、OR-05-7-2 Analyzing Interrupted In-Situ Tests on Vertical Boreholes Richard A. Beier, PhD Associate Member ASHRAE ABSTRACT The design of a geothermal heatpump system requires an estimate of soil thermal conductivity. An in-situ test on a bore- hole provides such an estimate, along with an estimate o
2、f the borehole resistance. Sometimes electrical power interrup- tions, running out offuel, or other equipmentproblems tempo- rarily disrupt the test andgreatly complicate the analysis of test data. This paper develops a method to estimate the elapsed testing time (or recovery time) when the efects o
3、f the inter- ruption dissipate suficiently so that the estimated thermal conductivity is changed by 10% or less. Afer the power is restored, the method can be used in the field to estimate the required recovery time. Because the test time using standard procedures can be prohibitively long following
4、 the interrup- tion, an analytical technique has been developed that shortens the minimum test time for a valid estimate of thermal conduc- tiviy. These methods are validated using data setsfiom a labo- ratory sandbox. Furthermore, the new method of analysis is able to estimate the soil thermal cond
5、uctivity from a data set that was previously not interpretable by standard (line-source) methods. INTRODUCTION A geothermal heat pump exchanges heat with the ground through a buried U-tube loop, where the heat transfer rate depends primarily on the thermal properties of the soil and borehole. An in-
6、situ test on a vertical ground loop may be performed to estimate some of these thermal properties, such as soil thermal conductivity. In the ideal test, heat is generated by an electric heater located at the surface to provide heat at a constant rate to the circulating fluid through the ground loop.
7、 At the same time, the inlet and outlet temperatures are measured for the circulating fluid. The average of these two Marvin D. Smith, PhD, PE Member ASHRAE temperatures is usually taken to represent the average in a vertical ground loop. This paper focuses on vertical ground In-situ borehole tests
8、are sometimes interrupted by elec- tric power outages or other unexpected events. In such cases, the length of the test prior to the interruption is often inade- quate to determine the value of soil thermal conductivity. If the test is restarted immediately after the power is restored, large swings
9、in the heat rate to the ground-loop complicate the analysis of the test. Most models assume a spatially uniform ground temperature at the start of the test. In cases where the field test is immediately restarted after an interruption, this assumption of uniform ground temperature is often invalid at
10、 the time of restart. Little guidance is available in the technical literature for handling interrupted tests. After a complete 48-hour test has been conducted, Martin and Kavanaugh (2002) recommend a 1 O- to 12-day waiting period before retesting a borehole. The waiting period allows the heat to di
11、ssipate around the borehole as the nearby ground temperature approaches the undisturbed temperature. If the initial test was shorter, they suggest the waiting period can be reduced in proportion to the reduced test time (Kavanaugh et al. 2001). However, such delays cost money when the equipment is a
12、lready on site and ready to go. If the interruption is short enough, the most expedient approach may be to resume the test when power is restored. The case of a short interruption has not been addressed adequately in the technical literature. Consider the temperature curve (open symbols) in Figure 1
13、, which is taken from a test in a laboratory sandbox with a two-hour interruption to the supplied electrical power. The heat input from an electrical heater in Figure 2 illustrates the power interruption between 9 and 11 hours. The temperature loops. * Richard A. Beier is associate professor ofmecha
14、nical engineering technology and Marvin D. Smith is professor and director of GHP research in the Division of Engineering Technology, Oklahoma State University, Stillwater. 702 02005 ASHRAE. 30 r Y a ci : 20 E Q l- U .- w 10 - m E z O Uninterrupted Test 0 Interrupted Test 0.1 1 10 10 Time (h) 16.7 s
15、 Y h 2500 L c 3 2000 m U 1500 O O O Q, 500 1000 .- L .CI i O 0.1 1 10 1 O0 Time (h) 737 586 v 8 n 440 g o 293 *= o Q 147 w, +ir O Figure 1 Loop temperature curves (normalized) from an laboratory sandbox. Figure 2 Electric power to heater during interrupted test #1. uninterrupted test and interrupted
16、 test #I in rise in a previous test without any interruption (with the same sandbox setup) is given by the solid symbols in Figure 1. The interrupted temperature rise (open symbols) eventually over- lays on the uninterrupted test curve (solid symbols). The late- time slopes are nearly the same, whic
17、h give comparable esti- mates of soil thermal conductivity. For this two-hour interrup- tion, a reasonable approach is to resume the test as soon as the power is restored. Cumulative test time, including the inter- ruption period, is 5 1 hours. Therefore, in some cases, restart- ing the test immedia
18、tely after power is restored makes sense. The tests in Figure 1 have been performed in a laboratory sandbox with dimensions of 6 fi x 6 ft x 60 ft (1.8 m x 1.8 m x 18 m). A 5 in. (O. 13 m) inner diameter aluminum pipe repre- sents the borehole wall and is centered along the length of the sandbox, wh
19、ich is horizontal. To simulate a borehole, a U-tube is placed inside the aluminum pipe. Bentonite grout fills the space between the U-tube outer walls and the inner wall of the aluminum pipe. Even though efforts were made to use the same heat rate during both tests, the heat rate cannot be repro- du
20、ced exactly. Because the heat input rates are different between the two tests in Figure 1, the temperature rise (T-Tnt) for the interrupted test has been multiplied by the ratio of the heat rates in the two tests, quninterntpted /qinterrupted With this adjustment, the curves should overlay except fo
21、r the effects due to the power interruption. Previous methods handle variable heat rates to geother- mal boreholes, but none of these methods has been applied to an interrupted test where the heat rate goes to zero. For instance, Beier and Smith (2003a) applied a deconvolution method to variable-rat
22、e tests, but this method requires a complete temperature and heat rate data set, even during the interrupted period. But during a power interruption, all temperature data may be lost. Shonder and Beck (1 999,2000) and Yavuzturk et al. (1 999) have applied numerical methods to analyze variable-rate t
23、ests, but they have not addressed inter- rupted tests. Because interrupted borehole tests are complicated to interpret, more than one approach has merit. Therefore, we apply several models to the problem in this paper. Our philos- ophy is to first present a detailed composite model with many paramet
24、ers that attempts to capture all the mechanisms of the interrupted test. The use of the detailed model provides a way to identify the most important parameters and develop simpler (line-source) models, which are easier to apply. The borehole test is an inverse problem where we seek the values of the
25、 parameters required by the detailed model. Because simpler models have fewer parameters, they tend to be more agile as they jump over the hurdles encountered in inverse problems. Of course, the simpler models must account for the dominant mechanisms if they are to give valid estimates. The detailed
26、 model provides a way to test the simpler models and ensure the simpler models include the dominant mechanisms. The present paper presents a simple method to estimate the recovery time for a restarted test after apower interruption. By recovery time, we mean the elapsed test time when the slope of i
27、nterrupted temperature curve in Figure 1 approaches the slope of the uninterrupted test curve to within 10%. Because the soil thermal conductivity is inversely propor- tional to the late-time slope, the corresponding error in the estimated soil thermal conductivity is also within 10%. Calcu- lations
28、 by Kavanaugh (2000) indicate a IO% error in soil ther- mal conductivity and diffusivity results in a 4.5% to 5.8% error in the design borehole length and a 1% change in the cooling capacity of a geothermal heat pump system. Thus, 10% error due to the interruption in the estimated soil thermal condu
29、ctivity is a reasonable target for the recovery time. In addition, a method of analysis is introduced to shorten the required testing time for such an interrupted test. With this ASHRAE Transactions: Symposia 703 Actual Borehole i o) - Model Fluid Soil o “ * “ aalJ O 0.1 1 10 1 O0 Time (h) Figure 3
30、(a) Geometry of actual borehole; (a) composite model of borehole. Figure 4 Loop-temperature rise measured in uninterrupted test in sandbox and temperature curve calculated by composite model. method one may be able to estimate soil thermal conductivity from some tests that were previously not interp
31、retable. All of these methods can be applied in the field to estimate the required testing time once the power is restored. estimates the soil thermal conductivity as 1.70 Btu/h.ft.“F (2.94 W/m.OC), while independent measurements with a heat probe give 1.63 Btu/h.ft.“F (2.82 W/m.“C). Forborehole res
32、is- tance, the model estimate is 0.263 hft.“F/Btu (O. 152 m.OC/w) COMPOSITE MODEL FOR INTERRUPTED TEST Before describing the simpler methods, we describe a detailed mathematical model for the interrupted borehole test and veri this model against the sandbox data in Figure 1. This sophisticated model
33、 provides a multitude of cases for testing the simpler methods, which are the main product ofthis paper. We apply a composite model that has been used in earlier studies (Beier and Smith, 2003b) to handle rate changes through superposition. In the composite model, the actual borehole geometry (Figur
34、e 3a) is represented by a simplified, radially symmetric geometry in Figure 3b. As in the models by Shonder and Beck (1999,2000) and Gu and ONeal(1995), the U-tube is replaced by a single pipe with an effective radius, rp. The soil and grout regions are treated as two distinct, inter- nally homogeno
35、us regions. compared with 0.299 h.ft.“F/Btu (0.173 m“CAN) from inde- pendent measurements. The composite model has been developed in terms of dimensionless groups to identi the least number of indepen- dent parameters. The dimensionless variables and parameters are: Dimensionless temperature Dimensi
36、onless radius The composite model does not explicitly account for the thermal resistance due to the pipe wall or the thermal resis- tance between the fluid and inside wall of the pipe. Similarly, no explicit account is made for any contact resistances at the these resistances are implicitly rolled i
37、nto the estimated value ofrp, which is directly related to the borehole resistance (Beier and Smith 2003b). P Fourier number (or dimensionless time) ks* Fo, = - (PCp)srp 2 pipe/grout interface or the grout/soil interface. Instead, all of Ratio of grout and soil thermal conductivities An analytical s
38、olution for the geometry in Figure 3b has been presented by Beier and Smith (2003b) in terms of dimen- (3) (4) sionless parameters. The composite model matches the loop temperature curve, as shown in Figure 4, for the uninterrupted test in the sandbox. As reported earlier (Beier and Smith Ratio of g
39、rout and soil volumetric heat capacities (5) (PC,) (PC,), 2003b), the models estimates of soil and borehole properties H=L compare well with independent measurements. The model 704 ASH RAE Transactions: Symposia b t, t;! Time Figure 5 Variations in heat rate represented as discrete step changes duri
40、ng an interrupted test. Dimensionless fluid-loop thermal storage Dimensionless borehole radius or borehole thermal resis- tance eXP(2zkgRb) (7) b rD,b = - = P Here the subscripts s and g denote soil and grout, respec- tively. The volume of the circulating fluid, Vr; in the fluid-loop storage paramet
41、er (Equation 6) is taken to be the volume ( Vr= 2 ri: L) based on the inside pipe diameter, Tin, and bore- hole length, L. See the nomenclature section for a list of all symbols. To develop a model for the interrupted test, we use the rate changes shown in Figure 5. Consider a borehole test with con
42、stant heat rate, qi, prior to time, ti, but the electric power supply is interrupted, and the rate suddenly goes to zero at time ti, as illustrated in Figure 5. Finally, the power is restored at time t2, and the heat rate is restored to q3, which may in general be at a different value than the initi
43、al rate, qi. For the interrupted test, superposition may be used to take into account these rate changes. To get the temperature response corresponding to the rates inFigure 5, one applies the constant-rate solution for each step rate change (qi - qi-i), which occurs at time, ti-i, and sums the cons
44、tant-rate solutions. The resulting dimensionless temperature, T, is expressed in terms of the dimensionless, constant-rate temperature response, TD,u, as where the Fourier number (dimensionless time), Fo, satisfies 90 32.2 s E Lo i- c n t 0%- 29.4 2 J Q) $! 85 w 3 al a E 80 26.7 +i, 75 23.9 1 10 1 O
45、0 Time (hr) Figure 6 Line-source and composite modeljts to the loop temperature curve from interrupted test #I in sandbox. the relationship, Fon-l I Fo 5 Fon, and qo = O at to = O. The Fourier number corresponds to time ti-l. Although the reference heat rate, qref) is arbitrary, in this paper the re
46、ference value is set to the last heat rate q3. For the composite model, T,u is evaluated from Equation A-2 in Beier and Smith (2003b). When the composite model constant-rate solution, TD,u, is substituted into Equation 8, the resulting loop temperature curve matches the temperature data from the int
47、errupted sand- box test in Figure 6. The scatter in the data at two hours is suspected of being due to some unwanted fluctuations in the pump rate for the circulating fluid through the loop. We focus on the temperature curve around the two-hour interruption period, starting at nine hours. The model
48、matches the rise of the loop temperature after power is resumed at 11 hours and the eventual recovery to the uninterrupted temperature curve. Thus, the composite model matches the temperature data collected throughout the interrupted test. LINE-SOURCE MODEL FOR INTERRUPTED TEST Although the composit
49、e model matches the interrupted test data, we seek a simpler model with only the dominant parameters. A simple model for the borehole is representing it as a line source of heat. Then, the temperature response for a constant heat rate from this line source can be substituted for T,u in Equation 8, and the superposition process can be carried out. The line-source model (Witte et al. 2002; Carslaw and Jaeger 1959; Beier and Smith 2002) for a single constant heat rate is given by ASH RAE Transactions: Symposia 705 where y is a constant approximately equal to 1.
