Estimation of Layered Ground Thermal Properties for Deep Coaxial Ground Heat Exchanger

Abstract

A ground heat exchanger (GHE) can efficiently exploit geothermal energy, and a ground source heat pump (GSHP) is an important type of geothermal application. The distributed thermal response test (DTRT) is widely used to measure layered ground thermal properties for shallow GHEs, but nowadays, there is a lack of studies applying the DTRT to deep coaxial GHEs (DCGHEs). This study proposes a new parameter estimation method (PEM) by adopting the DTRT data of a DCGHE to estimate layered ground thermal properties and applies the proposed PEM to simulated DTRTs under different boundary conditions, and the estimated values of the layered ground thermal properties are compared with the true values. Under heat output rate or inlet temperature boundary conditions, the relative errors of the thermal conductivities and heat capacities of ground estimated using the proposed PEM are basically within 2% and 4%, respectively, except for shallower layers with a depth range of 0–800 m. The larger errors for shallower layers may be caused by weaker heat transfer between the fluid and ground, and the errors are basically lower for higher heat output rates. The predicted fluid temperature distributions during 120 d using the estimated values of the layered ground thermal properties match well with those using the true values. The results show that the proposed PEM is viable for DCGHE DTRT interpretation under heat output rate and inlet temperature boundary conditions, is a cost-effective way to establish key parameters for GSHP design, and would promote geothermal development.

1. Introduction

Geothermal energy is an important renewable resource that can promote energy conservation and CO₂ emission reduction [1,2]. As a typical method of exploiting geothermal energy, a ground source heat pump (GSHP) is an efficient and environmentally friendly renewable energy technology and is becoming more and more popular under the background of global warming [1,3]. The core component of a GSHP system is the ground heat exchanger (GHE), which can be classified into shallow GHEs and deep GHEs (DGHEs). A DGHE has a borehole depth of at least 1000 m, resulting in higher initial investment costs. Thus, it is necessary to accurately evaluate the ground thermal properties to design a cost-effective GSHP. There are several methods to measure the ground thermal properties of a GHE, and the most cost-effective method is the thermal response test (TRT) [4].

TRT models the actual heat transfer process between the circulating fluid and ground and measures the time-varying parameters (including fluid temperatures, flow rate, heating power, etc.), and then the ground thermal properties are derived by comparing the test data with the heat transfer model. For a shallow GHE with a typical U-type configuration, the traditional TRT has been demonstrated to have high measurement accuracy [5]. For a shallow GHE with a coaxial-type configuration, the traditional TRT may have large errors because it regards the mean value of the inlet and outlet temperatures as the average fluid temperature; however, this assumption is not suitable for coaxial-type configurations [6,7]. Additionally, the traditional TRT only measures the inlet and outlet temperatures and regards the ground as a whole by neglecting the effects of layered ground thermal properties; therefore, the traditional TRT has some limitations. In fact, some studies have found that layered ground thermal properties have great effects on GHE performance and that the assumption of homogeneous ground may result in some errors [8,9].

To improve the measurement accuracy of the TRT, some researchers have proposed a distributed TRT (DTRT) to measure the depth-resolved fluid temperatures and obtain the distribution of ground thermal properties at different depths [10,11]. For a shallow GHE with a U-type configuration, a direct method is normally adopted for parameter estimation in the TRT, and the estimation process of ground thermal properties for each layer of ground in the DTRT is similar to that for the whole ground in the traditional TRT and the difference between them is as follows: the traditional TRT adopts the mean value of the inlet and outlet temperatures for estimation, and the DTRT adopts the mean value of the fluid temperatures in each layer of ground for estimation [12,13,14]. For a shallow GHE with a coaxial-type configuration, Nian et al. [15] adopted a genetic algorithm to estimate the layered ground thermal conductivities based on DTRT data and a 2D model. All the above studies show that the DTRT is more reliable and can offer more details about the ground.

There are a large number of studies that use the TRT to investigate shallow GHEs, but there are only a few studies that use the TRT for DGHEs [16,17,18,19]. Morchio et al. [16] adopted the traditional infinite line-source model (ILSM) to estimate the ground thermal conductivity of a DGHE, but the estimation result may have had large errors, mainly caused by the low precision of the ILSM. Beier [17] developed an accurate heat transfer model of a DGHE by taking the geothermal gradient into account, and when using the model for TRT data interpretation, the estimation precision was higher. Beier et al. [18] also studied the effects of layered ground thermal properties and found that adopting traditional 1D models for TRT data interpretation leads to large errors. Our previous study also found that groundwater seepage had great effects on the TRT result of a DGHE [19].

For the most widely used DGHE, i.e., the deep coaxial GHE (DCGHE), the borehole is deeper, and the ground thermal properties probably vary more greatly with depth. However, there is a lack of studies estimating the layered ground thermal properties of DCGHEs. Therefore, this study proposes a new parameter estimation method (PEM) to estimate the layered ground thermal properties of a DCGHE based on DTRT data; the proposed PEM is applied to a simulated DTRT of a DCGHE under a heat output rate boundary condition and an inlet temperature boundary condition, and the estimated layered ground thermal properties are compared with the true values.

The novelty of this study lies in its proposition of a new method to accurately measure the layered ground thermal properties of a DCGHE, which would enable the establishment of key parameters of DCGHE design and promote the development of GSHPs.

2. New PEM for Estimating Layered Ground Thermal Properties of DCGHEs

According to the geological conditions and measuring point distribution of a DTRT, the DCGHE can be divided into several layers. For any k-th layer of the DCGHE, it is assumed that the ground is homogeneous; the new PEM adopts a semi-analytical model to match the DTRT data to estimate the ground thermal properties, and an objective function is proposed to evaluate the matching. The semi-analytical model, objective function, and estimation procedure are introduced as follows.

2.1. Semi-Analytical Model for the k-th Layer of the DCGHE

In our previous study, a semi-analytical model was developed for a DCGHE [20], which adopts an analytical model to analyze the transient heat transfer in the ground and grout and adopts a numerical method to model the transient heat transfer in the fluids. Major assumptions of this model include the following: (1) for the ground and grout, vertical heat transfer is ignored, and only radial heat transfer is analyzed by the G function of a composite medium model and Duhamel’s theorem; (2) for the fluids, 1D steady flow and 1D transient heat transfer are assumed, and the convective heat transfer is calculated by the Gnielinski correlation. This study applies the semi-analytical model to each layer of the DCGHE, and therefore, only the initial and boundary conditions are different from those in ref. [20]. For any k-th layer of the DCGHE shown in Figure 1, the semi-analytical model is developed as follows.

Equations of the model are as follows:

$$\left[\pi r_{\mathrm{i}}^2{C}_{\mathrm{f}}+\pi (r_{\mathrm{o}}^2-r_{\mathrm{i}}^2)C_{\mathrm{ip}}\right]\frac{\partial T_{\mathrm{i}}}{\partial t}=m{c}_{\mathrm{f}}\frac{\partial T_{\mathrm{i}}}{\partial z}+\frac{T_{\mathrm{a}}-T_{\mathrm{i}}}{R_{\mathrm{ia}}},\left(Z_{\mathrm{k}-1}\le z\le Z_{\mathrm{k}},t>0\right)$$

(1)

$$\left[\pi (R_{\mathrm{i}}^2-r_{\mathrm{o}}^2)C_{\mathrm{f}}+\pi (R_{\mathrm{o}}^2-R_{\mathrm{i}}^2)C_{\mathrm{op}}\right]\frac{\partial T_{\mathrm{a}}}{\partial t}=-m{c}_{\mathrm{f}}\frac{\partial T_{\mathrm{a}}}{\partial z}+\frac{T_{\mathrm{i}}-T_{\mathrm{a}}}{R_{\mathrm{ia}}}-q,\left(Z_{\mathrm{k}-1}\le z\le Z_{\mathrm{k}},t>0\right)$$

(2)

$$q=\frac{T_{\mathrm{a}}-T_{\mathrm{eo}}}{R_{\mathrm{ae}}},\left(Z_{\mathrm{k}-1}\le z\le Z_{\mathrm{k}},t>0\right)$$

(3)

$$T_{\mathrm{eo}}^n=T_0+{\displaystyle \sum _{j=1}^n\frac{q^j-q^{j-1}}{K_{\mathrm{g}}}G(t^n-t^{j-1})},\left(Z_{\mathrm{k}-1}\le z\le Z_{\mathrm{k}},t>0\right)$$

(4)

$$T_{\mathrm{i}}=T_{\mathrm{a}}=T_0=T_{\mathrm{sur}}+az,\left(Z_{\mathrm{k}-1}\le z\le Z_{\mathrm{k}},t=0\right)$$

(5)

$$T_{\mathrm{a}}^{\mathrm{n}}=T_{\mathrm{i}}^{\mathrm{n}}+P_{\mathrm{a},\mathrm{k},1}^{\mathrm{n}}-P_{\mathrm{i},\mathrm{k},1}^{\mathrm{n}},\left(z=Z_{\mathrm{k}-1},t=t_{\mathrm{n}}\right)$$

(6)

$$T_{\mathrm{i}}^{\mathrm{n}}=T_{\mathrm{a}}^{\mathrm{n}}+P_{\mathrm{i},\mathrm{k},\mathrm{M}}^{\mathrm{n}}-P_{\mathrm{a},\mathrm{k},\mathrm{M}}^{\mathrm{n}},\left(z=Z_{\mathrm{k}},t=t_{\mathrm{n}}\right)$$

(7)

$$R_{\mathrm{ia}}=\frac{1}{2\pi r_{\mathrm{i}}{h}_{\mathrm{i}}}+\frac{1}{2\pi K_{\mathrm{ip}}}\ln \frac{r_{\mathrm{o}}}{r_{\mathrm{i}}}+\frac{1}{2\pi r_{\mathrm{o}}{h}_{\mathrm{a}}}$$

(8)

$$R_{\mathrm{ae}}=\frac{1}{2\pi R_{\mathrm{i}}{h}_{\mathrm{a}}}+\frac{1}{2\pi K_{\mathrm{op}}}\ln \frac{R_{\mathrm{o}}}{R_{\mathrm{i}}}$$

(9)

Equations (1) and (2) are the energy equations of the inner and annular fluids, respectively; Equation (3) is the heat transfer equation between the annular fluid and the outer wall of the outer pipe, Equation (4) is the heat transfer equation of the ground and grout based on the G function of a composite medium model and Duhamel’s theorem, Equation (5) represents the initial condition, Equations (6) and (7) represent the boundary conditions, and Equations (8) and (9) are the expressions of the thermal resistances. The above equations could be discretized and solved to compute the fluid temperature distributions. The above model is different from that in our previous study [20], and the novelty of the above model is that it is only used for the simulation of a layer of the DCGHE and that it adopts the experimental temperature data of the DTRT to set up the boundary conditions, i.e., Equations (6) and (7). At the same time, the proposed model also has the advantages of the model in our previous study, i.e., high precision, convenient modeling, and short computation time.

2.2. Objective Function

To evaluate the matching between the DTRT data and the proposed model, the root mean square error of the measuring point temperatures is selected as the objective function, as follows:

$$F_{\mathrm{k}}=\sqrt{\frac{1}{2M\left(N-s+1\right)}{\displaystyle \sum _{n=s}^N{\displaystyle \sum _{j=1}^M\left[{\left(T_{\mathrm{i},\mathrm{k},\mathrm{j}}^{\mathrm{n}}-P_{\mathrm{i},\mathrm{k},\mathrm{j}}^{\mathrm{n}}\right)}^2+{\left(T_{\mathrm{a},\mathrm{k},\mathrm{j}}^{\mathrm{n}}-P_{\mathrm{a},\mathrm{k},\mathrm{j}}^{\mathrm{n}}\right)}^2\right]}}}$$

(10)

where F_k indicates the objective function of the k-th layer of the DCGHE; N means the total number of testing times in the DTRT; s is the number of testing times equaling 10 h, which also means that the first 10 h of data are not considered in the matching.

When F_k is smaller, the matching between the calculated and experimental fluid temperatures is better. So, the ground thermal properties of the k-th layer of the DCGHE can be computed by searching the minimized F_k.

2.3. Estimation Procedure

The estimation procedure of the new PEM is as follows:

(1)

Divide the DCGHE into S layers. For each layer, establish the semi-analytical model and objective function, and then complete the following steps (2)–(8);

(2)

For any k-th layer of the DCGHE, firstly, determine the ranges of the ground thermal conductivity (K_s,k) and ground heat capacity (C_s,k), which are usually selected to be as wide as possible;

(3)

In the range of C_s,k, assign an initial value to C_s,k, which is regarded as the optimal C_s,k;

(4)

In the range of K_s,k, generate X random values based on the Monte Carlo method, which can be completed by calling the related function in the Fortran program or other software. It is worth noting that X is the number of the random values;

(5)

Input the optimal C_s,k and X random values of K_s,k into the semi-analytical model to compute the fluid temperature distributions and F_k, respectively; different F_k are compared to obtain the minimized F_k, and the corresponding random value of K_s,k is regarded as the optimal K_s,k;

(6)

In the range of C_s,k, generate X random values based on the Monte Carlo method;

(7)

Input the optimal K_s,k and X random values of C_s,k into the semi-analytical model to compute the fluid temperature distributions and F_k, respectively, and the random value of C_s,k corresponding to the minimized F_k is regarded as the optimal C_s,k;

(8)

Calculate the difference between the values of the optimal C_s,k in the adjacent iterations, and judge whether the difference is less than the setting value: if yes, go to the next step; if no, return to step (4);

(9)

Output the optimal ground thermal properties of all the layers.

Through the above steps, which are shown in Figure 2, the thermal conductivities and heat capacities of all the layers of the ground can be estimated.

3. DTRT Simulation

Because of a shortage of experimental data on the DTRT, this study conducts a DTRT simulation on a DCGHE based on a 3D numerical model [19], and the simulated DTRT data are used to validate the proposed new PEM.

3.1. Three-Dimensional Numerical Model

The 3D numerical model was developed in our previous study [19], and the main assumptions are made as follows:

(1)

The geometric structure of the DCGHE is simplified: the inner and outer pipes are assumed to have the same length, and the inlet temperature of the inner fluid is assumed to be equal to the average outlet temperature of the annular fluid;

(2)

The fluids are incompressible, and the standard k-ε model is used to analyze the turbulent flow of the fluids;

(3)

The thermal properties of all the materials are constant, and the ground thermal properties are homogenous in each layer of the ground.

Apart from the turbulent flow and heat transfer of the fluids, the 3D numerical model also simulates the heat conduction in the pipes, grout, and ground. In addition, the User Defined Function is applied to set the initial and boundary conditions, including the layered ground thermal properties. Meanwhile, the 3D numerical model outputs the time-varying fluid temperature distributions along the depth.

To validate the feasibility of the 3D numerical model to analyze the fluid temperature distributions with layered ground thermal properties, it is compared with the semi-analytical model [20]. A DCGHE is studied, the parameters of which are presented in

Table 1. Parameters of the DTRT of the DCGHE.

Parameter	Symbol	Value
DCGHE length	H	2000 m
Borehole radius	rb	155.5 mm
Inner radius of the inner pipe	ri	50 mm
Outer radius of the inner pipe	ro	55 mm
Inner radius of the outer pipe	Ri	84.3 mm
Outer radius of the outer pipe	Ro	88.9 mm
Thermal conductivity of the inner pipe	Kip	0.05 W/(m·K)
Heat capacities of the inner and outer pipes	Cip, Cop	2.2 × 106 J/(m3·K)
Thermal conductivity of the outer pipe	Kop	10 W/(m·K)
Grout thermal conductivity	Kg	4.0 W/(m·K)
Grout heat capacity	Cg	2.7 × 106 J/(m3·K)
Fluid thermal conductivity	Kf	0.6 W/(m·K)
Specific heat capacity of the fluid	cf	4200 J/(kg·K)
Fluid heat capacity	Cf	4.2 × 106 J/(m3·K)
Fluid viscosity	μf	1.14 × 10−3 kg/(m·K)
Geothermal gradient	a	0.03 K·m−1
Ground surface temperature	Tsur	293.15 K
Fluid mass flow rate	m	7.3 kg·s−1
Heat output rate	Q	200 kW

and

Table 2. Values of the layered ground thermal properties of the DCGHE [21].

Ground Layer	Depth (m)	Thermal Conductivity, W/(m·K)	Heat Capacity, J/(m3·K)
First layer	0–400	2.1	3.3 × 106
Second layer	400–800	2.5	3.0 × 106
Third layer	800–1200	3.6	2.9 × 106
Fourth layer	1200–1600	4.8	2.2 × 106
Fifth layer	1600–2000	5.5	2.1 × 106

[21], and there are five layers of the ground with different thermal properties. The meshing of the 3D numerical model is presented in Figure 3: because of symmetric geometry, half of the DCGHE is modeled; the meshing is conducted mainly along the radial, axial, and circumferential directions, and there are a total of 1,958,000 meshes and 2,116,876 nodes.

Figure 4 presents the comparison of the time-varying inlet and outlet temperatures of the two models. The differences between the two models are great early on but are less than 0.12 K after 10 h. It is worth noting that the differences during the first 2 h are very great and are not presented in Figure 4.

Figure 5 presents the comparison of the depth-varying fluid temperatures of the two models at different times. The annular fluid enters into the DCGHE at z = 0 m (z is the axial coordinate, i.e., depth), flows along the +z direction, and intersects with the inner fluid at z = H; the inner fluid flows along the −z direction. When z is small enough, the annular fluid temperature is higher than the ground temperature at infinity (i.e., the initial ground temperature), the annular fluid releases heat to the ground, and the annular fluid temperature decreases with z; when z is large enough, an inverse process would occur. As the inner fluid temperature is higher than the annular fluid temperature, heat is transferred from the inner fluid to the annular fluid, and the inner fluid temperature decreases along the flow direction. But, the inner pipe has a very low thermal conductivity and would restrain the heat transfer from the inner fluid to the annular fluid, and the inner fluid temperature varies very slowly with depth. At the time of 10 h, there are only a few differences between the fluid temperature distributions of the two models, and the minimum and maximum temperature differences are 0.04 K and 0.13 K, respectively. At the time of 80 h, their differences are also small and range from 0.01 K to 0.07 K. The result indicates that the 3D numerical model is feasible to simulate the depth-varying fluid temperatures.

In summary, it can be concluded that the 3D numerical model has high accuracy to simulate the DTRT of the DCGHE with the layered ground thermal properties.

3.2. Simulation of the DTRT of the DCGHE

The 3D numerical model is then used to simulate the DTRT of the DCGHE with the layered ground thermal properties. The information on the DTRT parameters and ground layers is shown in Table 1 and Table 2. The measuring points are arranged every 200 m, and there are a total of 22 measuring points to measure the fluid temperatures every 4 min. For the DCGHE, the common temperature boundary conditions include the heat output rate boundary condition and inlet temperature boundary condition. Because the temperature boundary condition may influence the precision of the DTRT of the DCGHE, both the two kinds of boundary conditions are simulated, respectively: the heat output rates of 100 kW, 200 kW, and 300 kW are simulated, respectively, and the inlet temperature of 293.15 K is also simulated. The simulated DTRT of the DCGHE is used to check the feasibility of the proposed PEM.

4. Results and Discussion

The simulated DTRT data are then input to the proposed PEM to estimate the thermal conductivities and heat capacities of all the layers of the ground, and the estimated values are compared with the true values to check the accuracy of the proposed PEM. The proposed PEM also divides the DCGHE into five layers, and each layer also has a length of 400 m. The ranges of K_s,k and C_s,k are selected as 1.0–7.0 W/(m·K) and 1.0 × 10⁶–5.0 × 10⁶ J/(m³·K), respectively. X is selected as 500, i.e., generating 500 random values of K_s,k or C_s,k in the proposed PEM.

4.1. Validation of the Proposed PEM under the Heat Output Rate Boundary Condition of the DTRT

The proposed PEM is applied to the simulated DTRT under the heat output rate boundary condition, and the estimated values of the layered ground thermal properties are compared with the true values, as shown in

Table 3. Estimated values and errors of the layered ground thermal properties based on the DTRT data with the heat output rate of 200 kW.

Ground Layer	True Ks,k, W/(m·K)	Estimated Ks,k, W/(m·K)	Relative Error of Ks,k	True Cs,k, J/(m3·K)	Estimated Cs,k, J/(m3·K)	Relative Error of Cs,k
First layer	2.1	2.06	1.8%	3.3 × 106	2.83 × 106	14.9%
Second layer	2.5	2.45	2.1%	3.0 × 106	2.94 × 106	3.4%
Third layer	3.6	3.56	1.0%	2.9 × 106	2.90 × 106	0.0%
Fourth layer	4.8	4.78	0.4%	2.2 × 106	2.17 × 106	1.9%
Fifth layer	5.5	5.49	0.1%	2.1 × 106	2.11 × 106	1.3%

and Figure 6. The relative errors of the estimated K_s,k are within 9%, and the majority of them are less than 2%; the relative errors of the estimated C_s,k are within 18%, and the majority of them are less than 4%. The errors of the estimated K_s,k are lower than those of the estimated C_s,k, which is because K_s,k is more sensitive to the fluid temperature distribution than C_s,k. The errors of the first and second ground layers are probably large, and the errors of the other ground layers are basically small enough. At the same time, the errors are basically smaller for a higher heat output rate, which may be explained by the fact that a higher heat output rate means larger fluid temperature variation, which leads to a smaller influence on the temperature measurement error.

The errors of the estimated ground thermal properties for different ground layers have large differences, which may be related to the fluid temperature distributions. Because the thermal conductivity of the inner pipe is small, the heat transfer between the annular and inner fluids is insignificant, and the heat transfer between the annular fluid and the ground is eminent. So, only the annular fluid temperature distribution at different times and heat output rates are depicted in Figure 7. As shown in Figure 7a with Q = 100 kW, for the depth range of 400–1200 m, the temperature difference between the annular fluid and ground is much smaller than that for other depth ranges, and the annular fluid temperature varies more slowly with depth and time, which means that the heat transfer between the annular fluid and ground is weakened. As shown in Figure 7b,c with Q = 200 kW and Q = 300 kW, for the depth range of 400–800 m, the temperature difference between the annular fluid and ground is much smaller than that for other depth ranges, the annular fluid temperature varies more slowly with depth, and heat may be transferred from the annular fluid to ground or vice versa. This is probably the reason why the thermal properties of the ground in the second layer are estimated inaccurately.

To check the accuracy of the estimated layered ground thermal properties to predict the fluid temperatures of the DCGHE, the estimated layered ground thermal properties are input to the 3D numerical model to simulate the fluid temperatures of the DCGHE, which are compared to those based on the true layered ground thermal properties, and the results are presented in Figure 8 and Figure 9. The outlet temperatures for the estimated layered ground thermal properties during 120 d match well with those for the true layered ground thermal properties for different Q, and the differences between them are less than 0.09 K. Their fluid temperature distributions also match well at t = 60 d and t = 120 d, and the differences between them are less than 0.09 K. The results show that the layered ground thermal properties estimated by the proposed PEM are accurate in predicting both the short-term and long-term fluid temperature distributions of the DCGHE.

4.2. Validation of the Proposed PEM under the Inlet Temperature Boundary Condition of the DTRT

The proposed PEM is also applied to the simulated DTRT under the inlet temperature boundary condition (the inlet temperature is 293.15 K), and the estimated values of the layered ground thermal properties are compared with the true values, as shown in

Table 4. Estimated values and errors of the layered ground thermal properties based on the DTRT data with the inlet temperature of 293.15 K.

Ground Layer	True Ks,k, W/(m·K)	Estimated Ks,k, W/(m·K)	Relative Error of Ks,k	True Cs,k, J/(m3·K)	Estimated Cs,k, J/(m3·K)	Relative Error of Cs,k
First layer	2.1	2.04	2.6%	3.3 × 106	3.05 × 106	8.1%
Second layer	2.5	2.44	2.4%	3.0 × 106	3.03 × 106	0.5%
Third layer	3.6	3.56	0.8%	2.9 × 106	2.90 × 106	0.4%
Fourth layer	4.8	4.76	0.9%	2.2 × 106	2.21 × 106	0.1%
Fifth layer	5.5	5.48	0.4%	2.1 × 106	2.13 × 106	0.3%

and Figure 10. The relative errors of the estimated K_s,k are less than 2.6%; the relative error of the estimated C_s,k is 8.1% for the first ground layer and is less than 0.5% for the other ground layers. The relative errors of the estimated results are basically lower for the deeper ground layer.

Obviously, the errors of the estimated ground thermal properties are very small except for the first ground layer, and to find out the reason, the annular fluid temperature distributions at different times are depicted, as shown in Figure 11. For the depth range of 0–400 m, the temperature difference between the annular fluid and ground is smaller than that for other depth ranges; the annular fluid temperature changes very slowly with depth and time. When the depth is deeper, the temperature difference between the annular fluid and the ground is larger, and heat transfer from the ground to the annular fluid is stronger. This is the reason why the thermal properties of the ground in the second layer are estimated inaccurately. The above result can be used to explain why the errors of the estimated results are large for the first ground layer and small for other ground layers.

As shown in Figure 7 and Figure 11, compared with the three heat output rate boundary conditions, the inlet temperature boundary condition shows lower inlet temperature and higher heat output rates during the first 80 h, which is why the estimated thermal properties based on the DTRT data under the inlet temperature boundary condition are basically more accurate than those under the heat output rate boundary condition.

The estimated layered ground thermal properties are also input to the 3D numerical model to simulate the fluid temperatures of the DCGHE, which are compared to those based on the true layered ground thermal properties, as shown in Figure 12 and Figure 13. The outlet temperatures for the estimated layered ground thermal properties match very well with those for the true layered ground thermal properties, and the differences between them are less than 0.06 K during the total 120 d. Their fluid temperature distributions also match very well at t = 60 d and t = 120 d, and the differences between them are less than 0.06 K. The results indicate that the layered ground thermal properties estimated by the proposed PEM under the inlet temperature boundary condition are also accurate in predicting the DCGHE performance.

5. Conclusions

The DCGHE is divided into several layers; a semi-analytical model and an objective function are developed for each layer, and then a new PEM is proposed, which adopts the DTRT data of the DCGHE to estimate the layered ground thermal properties. The proposed PEM is applied to the simulated DTRT of the DCGHE under different boundary conditions, and the estimated values of the layered ground thermal properties are compared with the true values. The main findings are as follows:

(1)

Under the heat output rate boundary condition, the relative errors of K_s,k and C_s,k estimated by the proposed PEM are within 9% and 18%, respectively, and the majority of them are less than 2% and 4%, respectively, and the errors are basically lower for higher heat output rate. Relative errors of the estimated ground thermal properties for shallower layers (i.e., depth range of 0–800 m for the studied case) are basically larger, which may be caused by weaker heat transfer between the fluid and ground. Meanwhile, when using the estimated values of the layered ground thermal properties for the DCGHE simulation, the predicted fluid temperature distributions during 120 d match well with those using the true values.

(2)

Under the inlet temperature boundary condition, relative errors of K_s,k and C_s,k estimated by the proposed PEM are within 3% and 9%, respectively, and the majority of them are less than 1%. Relative errors of estimated ground thermal properties for the shallowest layer with a depth range of 0–400 m are much larger than those for other layers, which is caused by weaker heat transfer between the fluid and ground. The predicted fluid temperature distributions during 120 d using the estimated values of layered ground thermal properties also match very well with those using true values.

The proposed PEM is validated, can be used for DCGHE DTRT interpretation under heat output rate and inlet temperature boundary conditions, and is cost-effective for offering key parameters of GSHP design, which would promote geothermal development. This study adopts a simulated DTRT to verify the proposed PEM, and in the future, field experiments of the DCGHE DTRT should be made, the accuracy of the proposed PEM should be checked further, and the optimization design of the DCGHE should also be investigated.