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Modeling approaches applied to the thermal response test: a critical review of the literature.


The geothermal thermal response test (GTRT, or simply TRT) is a fundamental and consolidated tool in the design of ground heat exchangers (GHEs), also called borehole heat exchangers (BHEs). Essentially, the TRT consists of either providing or extracting constant heating power to a carrier fluid that is circulated through an in situ pilot GHE, which consists of either coaxial tubes or a U-tube backfilled with a filling material and installed vertically in the ground. In practice, this is performed by adopting a transportable device generally arranged on a light trailer and equipped with temperature and flow rate sensors. The time-dependent behavior of the inlet and outlet mean temperatures of the fluid, suitably processed in order to take into account the fluid circulation time, represents the experimental data, providing an indirect measurement of the thermal response of the whole system when subjected to a constant heat flux injection or extraction condition. In its standard application, the analysis of these data constitute a solid procedure for the in situ estimation of the soil thermal conductivity, borehole resistance, and vertical temperature profile, i.e., of the important parameters for the design of ground coupled heat pumps (GCHPs; also referred to as ground source heat pumps [GSHPs]) and of the associated underground thermal energy storage (UTES) capacity (also referred to as borehole thermal energy storage [BTES]). Moreover, site geology and drilling conditions can be obtained from the test bore by providing other essential data for the design of the BHE as well as for the selection of drilling method and costing. A modified TRT methodology, in which the heat injection in the system is performed by dissipating a controlled amount of energy through electrical resistances placed inside the pipes of the GHE, was also recently suggested (Raymond et al. 2011a).

Review of the applications of this important alternative solution for space heating and cooling are found in Sanner et al. (2003), Florides and Kalogirou (2007), Yang et al. (2010), and Kim et al. (2010), the last with particular reference to field-scale evaluation.

The estimation of the unknown thermal properties is achieved through a comparison between the raw data for the time history of the average fluid temperature, experimentally acquired directly from the pilot BHE, and the corresponding values predicted by solving the partial differential equations (PDEs) governing the convection-conduction problem in the system. The identification of thermal properties using measured temperature values is a well-known inverse heat conduction problem (see, for instance, Beck et al. [1985] and Tervola [1989]). In general terms, within inverse heat transfer problems, the estimation procedure is generally constructed iteratively, with the aim of forcing a match between the theoretical predictions and the experimental results by tuning system properties. The amount of information that can be extracted by this parameter estimation procedure depends on the quality and quantity of the available experimental data and on the approximation adopted to derive the direct solution. In general, the methodologies adopted in literature to handle TRT data can be classified into two main approaches, which cannot be regarded as opposite, but complementary: analytical approaches that yield estimate of thermal properties directly without any iterative methods and numerical models that are generally implemented through iterative methods. In its original formulation, the TRT was performed by adopting a drastically simplified approximation in modeling the system thermal answer. The theoretical fundamentals of this simplified approach finds its origin in procedures developed to estimate thermal conductivity (Niven 1905), while the application to a borehole was developed much later by becoming a well-established reference solution for TRT in the 1980s.

This approach consists of approximating the BHE as an infinite line source releasing a constant heat flux, which is transferred to the soil. Under this assumption, known as the line source model (LSM), an analytical solution of the transient heat conduction problem in the soil is available (Carslaw and Jaeger 1959). The first TRT applications of this approach were presented by Mogensen (1983), and it has been widely applied afterward to determine the two main parameters necessary for the design of GCHPs, i.e., the soil thermal conductivity and the borehole thermal resistance (Eskilson 1987; Sanner et al. 2005). Other more complex analytical models have also been suggested, such as the cylinder source model (CSM), which takes into account the finite dimensions of the heat source (Kavanaugh and Rafferty 1997). An interesting approach to improve the design of TRT was presented by Raymond et al. (2011b), who observed that the methodologies adopted in standard pumping tests conducted in hydrogeology, where groundwater is pumped to perturb the hydraulic heads in an acquifer, can easily be transposed to the thermal problem. Raymond et al. (2011b), in fact, observed that in both problems, an inverse methodology is adopted to estimate system's properties by processing the system's response to a given perturbation.

Other approaches, more attractive because they allow for the extension of the predictive capability of the estimation procedure, have been developed by adopting the tools offered by numerical analysis. In fact, the possibility of deriving a numerical solution of the PDEs governing the phenomenon in principle allows for a more detailed description of the UTES behavior and for more information to be extracted by means of the parameter estimation procedure. A comparative test of both the LSM and a three-dimensional numerical approach, additionally accounting for groundwater flow effects, was presented and discussed by Raymond et al. (2011c). Their results, although derived under the direct rather than under the inverse problem approach, showed the necessity of adopting this advanced numerical approach to process TRT data to perform an optimal GCHP design.

Here, it must be remarked that the slenderness of the geometric domain in which the governing PDEs must be solved makes the numerical approach a very difficult challenge due to the extraordinary computational resources required (Al-Khoury et al. 2010). Therefore, most of the investigations reported in the literature in which the numerical simulation of both the long-term and the short-term behavior of ground loop heat exchangers under TRT conditions are presented are based on some kind of approximation, and they have mostly been developed by adopting a two-dimensional numerical schematization to estimate the same parameters generally considered within the LSM approach, i.e., the soil thermal conductivity and the borehole thermal resistance. Rarely have models been reported in which the governing equations are integrated in three dimensions, and even rarer are those investigations in which both the grout properties and the soil volumetric heat capacity are also estimated. Moreover, only in some cases has the whole set of equations describing the convection-conduction problem been considered.

In this review, the various modeling approaches applied to TRT data available in the literature are discussed and compared to point out their strengths and weaknesses in relation to the information that they are able to extract from the input data, as represented by the time history of the experimental fluid temperature.

This critical survey may provide important guidelines in the interpretation of TRT data and may be a useful tool facilitating the design of BHEs with respect to the energy analysis of GCHPs. In particular, the article intends to properly frame the TRT data processing procedure within the inverse heat transfer problem approach by providing a discussion about the different model's formulations reported in literature. In fact, the possibility of a more detailed modeling of the UTES potentially enables a more accurate estimation of the thermal properties of the soil and of the whole BHE, which constitutes a prerequisite for accurately sizing a complete geothermal system. As a case in point, it is observed that more accurate modeling may afford reductions in the time required to complete a TRT and then of the operational cost of the experimental procedure. Moreover, an accurate knowledge of the UTES behavior avoids useless over sizing of the borehole's depth by simultaneously assuring good performance for both heating and cooling systems based on GCHPs. On the other hand, it must be observed that a more detailed modeling does not necessarily imply that more accurate results are obtained. Moreover, most of the numerical approaches presented in literature to handle TRT data are often validated by means of comparison of the experimental fluid temperature data with the direct solution of the problem. This perspective disregards the relevant issue of the procedure, which is to get accurate values of the unknown grout and soil thermal conductivities and volumetric heat capacities under an inverse problem approach.

In the following, the PDEs governing the convective-conductive phenomena occurring in the BHE and in the UTES under the TRT conditions are first introduced, together with the fundamentals of the parameter estimation procedure. Both the single U-tube and the coaxial-tube BHE configurations are considered. The various approaches suggested in the literature for formulating approximate models are then illustrated. A model-based classification is adopted when introducing and reviewing the analytical and numerical one-, two-, and three-dimensional approaches available in the literature for processing experimental TRT results. The peculiarities and the weaknesses of each approach are discussed, with particular attention to the number of unknown parameters that can be estimated with each method. The analysis reported in the present review work demonstrates that an optimization of the modeling approach coupled with advanced parameter estimation procedures may result in a significant enhancement in the predictive capability of the TRT by maximizing the amount of information that can be extracted from the raw experimental data acquired in situ from the pilot BHE.

BHE geometry, governing equations, and parameter estimation procedure

BHE geometry

The most common TRT layout consists of either a single U-tube or coaxial tubes (generally a high-density polyethylene tube) installed vertically in a borehole backfilled with grout (generally a bentonite-concrete mixture) to ensure good thermal contact with the surrounding ground, which is assumed to be a homogeneous and isotropic medium. The carrier fluid (usually water) is circulated through the U-tube at a given flow rate, while in the coaxial case, the flow generally travels downward through the central pipe and comes back through the annulus. The wall thickness of the borehole is generally disregarded. The overall geometric layouts of these systems are illustrated in Figure 1 for the U-tube configuration and in Figure 2 for the coaxial-tube configuration.

In the following paragraphs, the governing equations for the two configurations are introduced, although it must be noted that the coaxial BHE is seldom encountered in TRT applications, whereas the U-tube layout is much more commonly found.


Governing equations

U-tube configuration

In the U-tube configuration, the system has a symmetry plane, and it can be considered practically unlimited in the radial direction, whereas in the axial direction, it is limited by two surfaces placed at z = 0 (the soil surface) and z = H (the depth of the heat exchanger), which can be assumed to be adiabatic. Under the assumption of a homogeneous and isotropic medium, the transient heat transfer conduction is governed by the Fourier equation:

C [partial derivative]T/[partial derivative]t = [lambda][[gradient].sup.2]T, (1)

where C and [lambda] are the medium volumetric heat capacity and thermal conductivity, respectively. In BHE applications, the cylindrical coordinate system (r,[theta],z) is generally adopted, and the Laplace operator [gradient] is then expressed as follows:


[[gradient].sup.2] = 1/r [partial derivative]/[partial derivative]r (r [partial derivative]/[partial derivative]r) + 1/[r.sup.2] ([[partial derivative].sup.2]/[partial derivative][[theta].sup.2] + [[partial derivative].sup.2]/[partial derivative][z.sup.2]. (2)

For the thermal problem schematized in Figure 1, Equation 1 should, in principle, be solved in each domain (i.e., soil, grout, and tube wall) with the initial condition

T([??], z, 0) = [T.sub.0,], (3)

where [??] indicates the position vector in the plane normal to z.

The coupling between the tube wall and the carrier fluid is suitably described by the Robin boundary condition:


where n is the outward normal to the internal surface of the tube wall, and the position vector [[??].sub.p] corresponds to the fluid-tube interface.

In Equation 4, h is the convective heat transfer coefficient associated with the working fluid flowing in the tube section with mean temperature [T.sub.f.]

For the convection problem, it is generally assumed that it is a one-dimensional phenomenon by disregarding the radial dependence of the fluid temperature. By neglecting the axial heat conduction into the fluid, the energy equation for the U-tube fluid flow is then

A[C.sub.f]([partial derivative][T.sub.f]/[partial derivative]t [+ or -] u [partial derivative][T.sub.f]/[partial derivative]z = h[[integral].sub.p] [T[([??].sub.p], z, t) - [T.sub.f](z,t)]dp, (5)

with the initial condition

[T.sub.f] (z, 0) = [T.sub.0]. (6)

In Equation 5, u is the fluid mean velocity in the axial direction, which has opposite signs for downward and upward fluid flow; A is the tube's cross-sectional area; [C.sub.f] is the fluid's volumetric heat capacity; and P is the tube inner perimeter.

The condition of constant power supplied to the working fluid may be suitably implemented by the periodic edge condition

[T.sub.f](inlet section) = [T.sub.f](outlet section) + [DELTA][T.sub.f,], (7)

where [DELTA][T.sub.f] is constant over the whole temporal domain.

The continuity condition for both temperature and heat flux at the interface between the solid domains of different thermal proprieties completes the statement of the problem. Moreover, within the one-dimensional approximation of the convection problem, the U-tube bend may be modeled by simply imposing the condition that the fluid temperatures on the two legs of the tube are equal at z = H.

The formulation of the above-reported PDEs, which must be solved simultaneously, completed by the boundary conditions, should be unique, although several different approximating models can be developed from these equations by disregarding some terms. Unfortunately, other ambiguous or unnecessarily complicated formulations of the equation set (Equations 1-7) are reported in literature. For instance, in Al-Khoury et al. (2010) and Gustafsson and Westerlund (2010), the definition of the boundary condition of the problem is somewhat ambiguous.

Coaxial-tube configuration

In the coaxial-tube configuration, the system has an axis of symmetry, and the problem is consequently two-dimensional. As in the U-tube configuration, it can be considered practically unlimited in the radial direction, whereas in the axial direction, it is limited by two surfaces located at z = 0 and z = H that have been assumed, according to the standard approach, to be adiabatic.

For the soil, grout, and tube wall domains, the Fourier equation (Equation 1), here formulated in an (r, z) coordinate system, holds with the initial condition given by Equation 3.

The coupling between the carrier fluid at temperature [] in the annular channel and the grout is realized by setting the Robin condition at the boundary surface:


where n is the outward normal coordinate to the internal surface of the external tube wall, and [h.sub.o] is the corresponding convective heat transfer coefficient associated with the annular channel flow.

As in the U-tube configuration, the convection problem is assumed to be a one-dimensional phenomenon by disregarding the radial dependence of the fluid temperature. By neglecting the axial heat conduction into the fluid, the energy equation for the inner flow having temperature [] is then

[A.sub.i][C.sub.f]([partial derivative][]/[partial derivative]t + [u.sub.i] [partial derivative][]/[partial derivative]z) = U[[](z, t) - [](z, t)], (9)

where U represents an overall heat transfer coefficient for a unit BHE length, accounting for both the internal and external convection and, if necessary, of the conductive thermal resistances of the tube's wall. Analogously, for the flow in the annular channel at a temperature [T.sub.o,] the energy equation is given by


In Equations 9 and 10, [u.sub.i] and [u.sub.o] are the fluid mean velocities in the axial direction in the inner pipe and the annular channel, respectively; [A.sub.i] and [A.sub.o] are the respective cross-sectional areas; and [P.sub.o] is the external perimeter of the annular channel. The condition of constant power supplied to the working fluid may be suitably implemented by the periodic edge condition given by Equation 7.

The convective heat transfer coefficients h in Equation 4, [h.sub.o] in Equation 8, and the convective coefficients embedded in the overall heat transfer coefficient U can be derived from the heat transfer correlations holding for the internal forced-convection problem. In case of fluids flowing under the turbulent regime, the most widely accepted correlation, regardless of the thermal boundary conditions, is the Dittus-Boelter correlation holding for turbulent pipe flow (Incropera and DeWitt 2002), which expresses the dimensionless heat transfer coefficient (Nusselt number, Nu) as a function of the Reynolds number, Re, and the Prandtl number, Pr, as follows:

Nu = 0.023 [Re.sup.4/5][Pr.sup.n], (11)

where the exponent n is equal to 0.4 and to 0.3 for heating and cooling conditions, respectively. Incase of fluids flowing under the laminar regime, the estimation of the convective heat transfer coefficient is more problematic due to the dependence of the Nusselt number on the thermal boundary condition and geometric configuration (Shah and London 1978).

The continuity condition for both temperature and heat flux at the interface between the grout and the soil domains completes the statement of the problem. Moreover, in the one-dimensional approximation of the convection problem, the fluid temperature of the inner and channel fluid flows are equal, z = H.

Parameter estimation procedure

The direct solution of the governing equations (Equations 1-10) provides the transient temperature field. This solution requires the knowledge of the system's thermal parameters together with the initial and far-field temperature conditions, borehole geometric parameters, and fluid flow conditions. In a parameter estimation approach, the problem may instead be inversely solved to estimate the system's thermal parameters by adopting as input data the temperature history of either the fluid or of some points located within the domain. The parameter estimation procedure can be constructed by forcing a matching of the experimental data with the values derived by the direct solution by tuning the systems' thermal properties (Beck et al. 1985). If the inverse heat transfer problem involves the estimation of only few unknown parameters from transient temperature measurements, as occurs in TRT applications, this matching may be suitably performed using a least squares minimization approach within an iterative scheme. A general and straightforward minimization methodology is represented by the Gauss linearization method (Beck and Arnold 1977), although other more complex solution strategies are available, such as the conjugate gradient method (Ozisik and Orlande 2000). In the standard TRT approach, the system thermal solution is derived from the average inlet and outlet mean fluid temperatures. Using the the Gauss linearization method, the estimation procedure algorithm can be then developed by minimizing the following function:

S(P) = [[Y - T(P)].sup.T] [Y - T(P)], (12)

where Y ([Y.sub.i] = measured temperature at time i, i = 1, ..., M) expresses the temperature data measured at the exit section of the BHE, and T(P) is the corresponding estimated temperature obtained from the solution of the direct problem using the current estimates of the unknown parameters P([P.sub.j] = unknown parameters, j = 1, ..., N). The iterative procedure of the Gauss linearization method for the minimization of the above norm S(P) is constructed by setting

[P.sup.k+1] = [P.sup.k] + [DELTA][P.sup.k,], (13)

where [DELTA][P.sup.k] is the vector of unknown parameter increments at iteration k obtained by minimizing the function S([P.sup.k+1]) with respect to [P.sup.k]. This is done by linearizing the temperature vector with a Taylor series expansion. The minimization of the function S(P) can thus be expressed as a system of N linear algebraic equations, where [DELTA][P.sup.k.sub.j] denotes the unknown increments. The solution of this equation system is given by

[DELTA][P.sup.k,] = [[[J.sup.k]].sup.t] [T([P.sup.k]) - Y]/[[[J.sup.k]].sup.t][[J.sup.k]], 14)

where [J.sup.k] is the sensitivity matrix, in which the jth component is defined as

[J.sup.k.sub.j] = [M.summation over (i=1) [partial derivative][T.sup.k.sub.i]/[partial derivative][P.sub.j]. (15)

The iterative procedure stops when the increment reaches a sufficiently small threshold value. When dealing with noisy data, a further stopping criterion based on the discrepancy principle (Alifanov 1994) may also be considered, in which the iterative procedure stops when the residuals between the noisy data and the estimated temperature are of the same order of magnitude as the measurement uncertainty.

By disregarding the tube wall thermal resistance, the model described by the above equations contains the following independent parameters: thermal conductivity and volumetric heat capacity of the soil and the grout ([[lambda].sub.s], [lambda].sub.g], [C.sub.s], and [C.sub.g], respectively), the geometric parameters (shank spacing and radius and length of the tubes), initial and far-field temperature value [T.sub.0], fluid flow conditions (fluid velocity and thermal properties together with the convective heat transfer coefficients), and the supplied heat load. For design purposes, the borehole thermal resistance [R.sub.b] is also often considered, as follows:

[R.sub.b] = [T.sub.f] - [T.sub.b,]/q, (16)

where [T.sub.f] is a reference fluid temperature, and [T.sub.b] is a reference temperature at the borehole wall.

The procedure expressed by Equations 12-15 can, in principle, be applied to TRT data (represented by the time history of the mean fluid temperature between the inlet and outlet tube sections) by considering as unknown parameters some or all of the above-listed variables. However, the estimation procedure implemented in the TRT is often formulated as a two-variable model, as outlined in Table 1, which summarizes the most significant scientific contributions available in the literature in which the inverse heat conduction problem approach is applied to the TRT.

In devising effective parameter estimation strategies, a sensitivity analysis can be of great help, together with an uncertainty analysis on the input raw data. In particular, a prior sensitivity analysis allows for an exploration of the parameter space by verifying that the concurrent estimation of the unknown variables is feasible. With respect to the TRT, several authors (Austin et al. 2000; Shonder and Beck 2000a, 2000b; Beier and Smith 2003) observed that the simultaneous estimation of both the soil and the grout thermal conductivities and volumetric heat capacity over the entire duration of the TRT is problematic because some of these parameters are strongly correlated. The sensitivity analysis results (Bozzoli et al. 2011) enabled instead the optimization of the minimization procedure, so that the four unknowns ([[lambda].sub.s], [lambda].sub.g], [C.sub.s], and [C.sub.g]) were successfully recovered by splitting the estimation procedure into two subsequent steps in which the short-term and long-term TRT data were considered separately.

Modeling approaches to TRT

Regarding the various TRT models that have been developed, both analytical and numerical, within either a one-dimensional or multidimensional spatial schematization, each of them is based on some kind of approximation that reflects a simplified formulation of the equation set (Equations 1-10).

Analytical approaches

The one-dimensional models on which the TRT estimation procedure relies are mostly derived by analytical approaches to the unsteady-state heat transfer problem in the UTES. The most important model is the LSM, based on the so-called Kelvin line source theory (Kelvin 1882; Carslaw and Jaeger 1959). This simple model represents the starting point for discussing the estimation procedure embedded in the TRT and applied to a vertical BHE. The LSM consists of the unsteady heat conduction problem in an isotropic homogeneous medium with constant properties in which an infinite line heat source is present. This model is obtained in practice by considering the BHE as lumped in an infinite line source located on the system axis. The transient Fourier equation (Equation 1) in the soil is then formulated by considering the only radial spatial dependence, and the heat injected into the system is expressed by a heat flux line boundary condition.

The analytical solution of the problem, expressing the temperature history at a given radial position in the soil, is given by


where q is the heating rate per unit length of the source, which is uniform and constant since the initial moment at t = 0; and [[lambda].sub.s] and [[alpha].sub.s] are the soil thermal conductivity and diffusivity, respectively.

For large time values, the solution can be simplified further. By introducing a thermal resistance, known as the borehole resistance [R.sub.b,] in order to take into account the effects of borehole geometry and thermal properties, and by further assuming a constant and uniform value [T.sub.0] for both the undisturbed (initial) ground and fluid temperature, this simplified model results in the following equation:

[T.sub.f](t) = [T.sub.0] + q[R.sub.b] + q/[pi][[lambda].sub.s] [1n(4[[alpha].sub.s]t/[r.sup.2.sub.b]) - [gamma], (18)

where [r.sub.B] is the borehole radius, and [gamma] is the Euler's constant. For the reference fluid temperature [T.sub.f], the arithmetic mean value between the inlet and outlet fluid temperature is generally considered, while for the system boundary, a constant and uniform temperature value equal to [T.sub.0] is assumed. With increasing time, the systematic error introduced by the approximation given by Equation 18 with respect to Equation 17 diminishes, and it becomes lower than 10% if t[[alpha].sub.s]/[r.sup.2] [greater than or equal to] 5.

The parameter estimation procedure applied to the TRT coupled to the LSM ends in a simple least squares fitting of the experimental data; for large time values, the thermal conductivity of the soil is determined as the slope of the straight line that represents the time evolution of the mean fluid temperature on a semilogarithmic scale.

Although this model is characterized by great simplicity and has been proven to be a robust tool in practical applications, as reported in the past by Mogensen (1983), Eskilson (1987), and Gehlin (1998), and more recently by Esen and Inalli (2009), Roth et al. (2004), and Sharqawy et al. (2009), when focusing on particular locations, it recovers information about the BHE and the UTES that cannot de facto be considered exhaustive for an accurate sizing of a GCHP system. In fact, this model enables the restoration of the values of two parameters that are important for the design of BHEs, i.e., the effective ground thermal conductivity and the borehole thermal resistance, which characterizes the coupling between the heat carrier fluid and the borehole wall. It does not, however, allow the determination of the volumetric heat capacity of the UTES and, therefore, its thermal diffusivity.

The soil thermal capacity is usually assumed to be known and constant; however, this property is generally variable by up to 100% for the same ground or rock type (ASHRAE 2007; Gillies et al. 1992). Even where the determination of thermal capacity is difficult because it is very sensitive to the borehole thermal resistance (Beck et al. 1956), it is a necessary property for the correct assessment of heat exchanger performance. Moreover, the LSM necessarily does not consider the u-tube structure, for which the description necessarily requires at least a two-dimensional schematization; therefore, the LSM disregards the effect of the properties of the filling material on system thermal response and borehole thermal resistance. other weaknesses of the TRT implemented with the LSM are due to the data-selection procedure, which must be tailored to the particular experimental conditions and geometric configuration of the BHE and to the choice of test duration. These aspects, often left to the experience and sensitivity of the operator, can greatly affect the accuracy of the estimated properties and also the cost of the test. The measurement time necessary to obtain sufficient data for a reliable analysis has been much discussed since the beginning of response test measurements. Austin et al. (2000) found that a test length of 50 h was satisfactory for typical borehole installations, whereas Gehlin (1998) recommended test lengths of approximately 60 h. Comparisons of tests of different durations were reported by Austin et al. (2000) and Witte et al. (2002), who also thoroughly discussed other caveats of the experimental procedure embedded in the TRT, such as the effect of fluctuations in electrical power, inaccurate measurements of the undisturbed ground temperature, and uncontrolled heat losses or gains to or from the environment due to insufficient thermal insulation. The results presented by Witte et al. (2002) are supported by the comparison between the values of the soil thermal conductivity obtained by means of the estimation procedure based on the TRT with the LSM and by means of independent laboratory measurements. A method to estimate the minimum test duration required to estimate the soil thermal conductivity to within 10% of its long-term estimate was also suggested by Beier and Smith (2003), who developed a solution of the one-dimensional form of the heat diffusion equation by using the Laplace transform method. Their results show that the minimum test duration may vary significantly, depending on the BHE geometry and UTES thermal properties. Another factor that may impair or limit the accuracy of the TRT is due to the presence of groundwater flow, which necessarily alters the thermal answer of the BHE with respect to the assumption of a medium with a purely conductive transport mechanism (Gehlin and Hellstrom 2003a). Gustafsson and Westerlund (2010) suggested a multiple-injection-rate TRT, also considered by Witte and van Gelder (2006) and by Witte (2007), to detect the presence of convective heat flow due to groundwater movement, and they examined the effect of this phenomenon on the estimated value of the ground thermal conductivity and borehole thermal resistance.

The superposition principle, implemented with the Duhamel theorem (Beck et al. 1985) coupled to the LSM, was adopted in order to extend the applicability of the TRT to experimental conditions affected by variable heat injection rates (Raymond et al. 2001b). Field tests that considered the test duration, as well as the test accuracy, were also discussed in Raymond et al. (2011b).

The conditions of the interrupted TRT was also considered by Beier and Smith (2005) and Beier (2011). In Beier (2011), the definition of an equivalent time, which enables the implicit taking into account of the discontinuity in the perturbation applied to the BHE, was suggested.

More complex analytical models have been suggested, such as the CSM (Carslaw and Jaeger 1959), which takes into account the finite dimensions of the heat source. The cylindrical source solution under the constraint of a constant heat flux is as follows:


where the Fourier number and the dimensionless radial coordinate are defined as follows: Fo = [[alpha].sub.s]t/[r.sup.2.sub.0], [eta] = r/[r.sub.0,], q is the heating rate per unit length, [r.sub.0] is the equivalent radius of the BHE, and J and Y are the Bessel functions of the first and second kind.

In particular, Kavanaugh and Rafferty (1997) suggested an iterative procedure that enables the estimation of both the effective thermal conductivity and diffusivity of the ground. The procedure is based on a prior estimation of the effective thermal resistance of the ground by means of a short-term TRT in which first-attempt values for [[lambda].sub.s] and [[alpha].sub.s] are adopted. Subsequently, the cylinder source solution is applied to update the values for [[lambda].sub.s] and [[alpha].sub.s] until the estimation procedure converges. The CSM was also adopted by Fujii et al. (2009), who applied this approach by considering the axial variation of the fluid temperature acquired experimentally by means of optical fiber thermometers. The experimental data were used to estimate the thermal conductivity distribution along the BHE depth by retrieving the information about the local geological and groundwater conditions.

With this regard, it must be observed that the possibility of adopting the axial temperature distribution as input data of the parameter estimation strategy, implemented by Equations 12-15, can help in extracting additional information with respect to the standard TRT approach, which was also discussed in Rohner at al. (2005) and Acuna (2010). Nevertheless, this technique, which can be regarded as innovative since it helps in retrieving not only the soil equivalent thermal conductivity but also information about the vertical geological stratification and/or groundwater occurrence of the site, requires additional experimental facilities that, at the moment, are rather expensive.

The finite length of the borehole has also been taken into account by Eskilson (1987), who presented a solution expressed in integral form by considering a combined analytical and numerical solution technique. Regarding the analytical approach to the TRT, an important advance was made by Bandos et al. (2009), who presented analytical expressions for the asymptotic behavior of the ground temperature for the intermediate and long time scales by accounting for the finite dimensions of the BHE, for the presence of a vertical temperature (geothermal) gradient and for the time dependence of the ambient temperature. Moreover, they discussed several approaches that can be adopted to appropriately define the reference fluid temperature to use in the estimation procedure. A generalization of this analytical model was presented in Bandos et al. (2010), which took into account the effect of anisotropic diffusion in a multilayered soil medium.

Wang et al. (2010) suggested an alternative TRT method based on a constant temperature condition that was demonstrated to have several advantages, such as the reduction of the time period necessary to reach steady-state system behavior with respect to the conventional TRT based on a constant heat flux condition. The analytical solutions describing the BHE behavior described so far did not consider the short-term response of GHEs, which necessarily depends on the thermal properties of the filling material and on the thermal inertia of the whole system. Several analytical solutions that consider the early regime of the transient behavior in the BHE are found in the literature, but they have rarely been adopted in estimation procedures under the TRT approach. The short-term response was recently addressed by Javed (2010), who discussed possible modeling approaches by presenting both analytical and numerical solutions for the short time-scale regime of the BHE behavior.

Lamarche and Beauchamp (2007) presented an analytical approach to the problem by deriving a solution in the closed form of the Fourier equation in a compound infinite medium defined by concentric cylinders under both constant heat flux and convection boundary conditions at the inner tube surface. The solution was compared to numerical results of both the same problem and of a real u-tube BHE configuration. The short-term response was also considered by Bandyopadhyay et al. (2008), who derived a solution in the Laplace domain of a simplified GHE model by taking into account the thermal capacity of the circulating fluid. These interesting analytical approaches, which enable the simulation of the so-called short-term response of the BHE that accounts for the grout thermal properties, has not been adopted in estimating BHE and UTES thermal properties under an inverse approach.

The analytical models described thus far neglected several aspects that may limit the predictive capability of the estimation method in assisting the GCHP design, although it is noted that these models provided reference points for further development.

Numerical models

More promising approaches to improve the predictive capabilities of the TRT can be found in the parameter estimation procedures supported by numerical tools. The main advantage of these models is that they allow a more accurate description of the system. Consequently, they either do not require approximations in the equations set (Equations 1-10), or, in some cases, they require approximations that are less severe than those of the LSM.

Numerical one-dimensional model

A model in which a spatially one-dimensional description is adopted to describe the UTES thermal response was validated by Shonder and Beck (2000a; 2000b). The finite-difference solution of the Fourier equation is obtained in the soil domain, and the fluid heat transfer in the pipes is regarded as a steady-state phenomenon by considering the heat capacity of the fluid as lumped into a "film." The presence of the grout and the geometric arrangement of the pipes are modeled as a single pipe of an effective radius. Under this approach, the Fourier equation (Equation 2) is solved by assuming that the temperature field is constant along the z-axis and also that it is invariant under spatial rotation about the same axis. The estimated values of the thermal conductivity of both grout and soil and the borehole resistance were in good agreement with independent measurements (Shonder and Beck 1999) and with values obtained by the LSM and CSM (Shonder and Beck, 2000a). The main benefits of this approach are that the model uses the field-measured power input data rather than an average value; moreover, the method provides statistical estimates of the confidence intervals for the unknown parameters. A one dimensional finite-difference numerical model was also presented by Gehlin and Hellstrom (2003b), who compared this approach with three different analytical modeling approaches for assisting the TRT estimation procedure. Their conclusions confirmed that the LSM is the fastest and simplest model that guarantees a good approximation if at least 50 h of measurement is considered. With respect to the numerical model, they observed that, as expected, the one-dimensional approach is unable to capture the short-term thermal response of the system.

Numerical two-dimensional model

Several approaches are available in literature in which the spatial dependence of the temperature field is described by adopting a two-coordinate system. Most of these methods adopt either a "horizontal slice" or a "vertical slice" approach. In the vertical slice approach, the heat diffusion equation is invariant under spatial rotation about the z-axis of the vertical BHE, and therefore, only the z and r dependences in Equation 2 are considered. Under the horizontal slice approach, the r and [phi] dependences of the temperature field are accounted for, but axial effects are disregarded. Both approaches represent approximations of the BHE if the U-tube configuration is considered; the vertical slice approach can correctly describe BHE behavior if the coaxial-tube configuration is modeled.

Wagner and Clauser (2005) coupled a parameter estimation procedure with a transient spatially two-dimensional thermal model under the vertical slice approach with the aim of recovering the soil thermal conductivity and heat capacity per unit volume. In this method, the minimum of the objective function, represented by the squared difference between the estimated and calculated data, is performed by adopting a simulated dataset and by graphically detecting the minimum misfit value. The vertical slice approach was also considered by Zanchini et al. (2010) when describing a coaxial BHE by focusing on the effect of thermal short circuiting on the performance of the heat exchanger. A pseudo-two-dimensional model was considered by Fujii et al. (2009), who applied the CSM to sublayers into which the ground was vertically subdivided. A similar approach, based on a distributed TRT performed by means of optical fiber temperature sensors, was also discussed by Acuna (2010).

Austin et al. (2000) adopted a two-dimensional modeling approach under the horizontal slice approach by integrating the governing PDEs with the finite-volume numerical method. The two-dimensional approach, also considered in Yavuzturk et al. (1999), consists of approximating the BHE geometry as a "pie sector" lying on a plane orthogonal to the BHE axis. By considering a non-symmetric distribution between the two legs of the U-tube, the convection resistance associated with the fluid flow is incorporated into the pipe's wall resistance, and a time-varying heat flux is assumed to enter through the pipe wall. This approximation is obtained by neglecting the axial dependence of the temperature field in Equation 1 and by simplifying the coupling between the tube wall and the carrier fluid with an equivalent thermal resistance.

Austin et al. (2000) validated this methodology with the aim of simultaneously estimating the thermal conductivities of both the soil and grout by comparing the results regarding a medium-scale laboratory experiment with independently measured values, and they observed a maximum deviation of approximately 2.1%. The problems involved in the estimation of parameters other than the soil and the grout thermal conductivity was discussed. Moreover, a sensitivity analysis was also presented to determine the influence on the estimated value of the ground thermal conductivity of the parameter values used as inputs in the estimation procedure. In particular, the repercussions on the estimated values of the soil thermal conductivity of uncertainties in the far-field temperature, volumetric specific heats, shank spacing, and borehole radius were considered, and the effect of the length of the test was also accounted. The total estimated uncertainty was in the range of 9.6%-11.2%. The work by Austin et al. (2000) hinted at the possibility of adopting a three-dimensional numerical approach to optimize the modeling of the early transient regime, which is strongly dependent on GHE geometry and grout thermal properties. Notably, the authors observed that huge computational resources would be required to achieve this aim.

Numerical three-dimensional model The GHE three-dimensional models available in the open TRT literature have been used mostly to generate synthetic TRT data, and only recently have these models been coupled with parameter estimation procedures applied to experimental data under an inverse problem approach. Marcotte and Pasquier (2008) built a three-dimensional model of a GHE in the COMSOL Multyphysics[R] environment; they did not integrate the energy transport equation for the U-tube fluid flow, but they modeled the fluid domain as a solid one by solving the Fourier equation under the assumption of an anisotropic fluid thermal conductivity tensor. Moreover, in their model, the convective resistance between the pipe wall and the fluid flow was neglected. The main outcome of this model was a new definition of the average fluid temperature to be employed in the usual estimation procedure based on the line source approach, which, thanks to this improvement, enabled a better estimation of the borehole thermal resistance. Signorelli et al. (2007) developed a three-dimensional GHE model in the FRACTure[R] environment that can accurately simulate the advective thermal transport in the U-tube using special one-dimensional tube elements surrounded by three-dimensional matrix elements. They implemented this model to generate synthetic TRT response data that enabled an evaluation of the effects of heterogeneous subsurface conditions and groundwater movement on the usual estimation procedure based on the line source approach. The model results were matched with experimental TRT data by tuning the soil thermal conductivity. Lamarche et al. (2010) implemented a complete three-dimensional model in the COMSOL Multyphysics1[R] environment in which fluid flow in the pipe was modeled as a one-dimensional problem using the classical advection equation. The thermal response of this model was processed by particularly discussing the estimation of the borehole resistance.

Bozzoli et al. (2011) presented a two-step parameter estimation procedure (TSPEP) based on a numerical three-dimensional model of the geothermal system. The procedure was applied to both simulated and experimental standard TRT data to recover the grout and soil thermal conductivities and volumetric heat capacities. The TSPEP algorithm essentially consists of two sequential procedures to be applied to the data for two different time intervals. Specifically, in the early transient regime, the parameter estimation procedure enables the recovery of the grout thermal conductivity and heat capacity per unit volume. These values are then used as input values in the second step, in which the parameter estimation procedure is applied to the late transient regime to recover the soil thermal conductivity and heat capacity per unit volume. To improve the accuracy of the procedure, further reiterations of these two steps were used. The time separation in the estimation of soil and grout properties partially decouples the two problems, making the estimation of these four parameters feasible. The selection of the time intervals in which the two steps are applied is a critical point of this multi-parameter estimation strategy. A preventive sensitivity analysis was shown to be the correct strategy to assist the parameter estimation procedure, together with the analysis of the residuals between the computed and the experimental data versus time. The results reported by Bozzoli et al. (2011) confirmed what Austin et al. (2000) pointed out; i.e., that three-dimensional modeling applied to a TRT can overcome the problems associated with the simultaneous estimation of both soil thermal conductivity and soil volumetric specific heat.


The modeling of the BHE under TRT conditions has undergone many improvements since it was first formulated. In the present review, the various modeling approaches suggested in literature for TRT analysis were discussed and compared to point out their strengths and weaknesses in relation to the information extracted from the input data, represented by the experimental temperature time history. In their initial formulation, these models were based on the LSM, which is a drastically simplified approximation of the BHE system. Although this standard procedure still represents a starting point that provides a simple and rapid tool for estimating the soil thermal conductivity and the borehole thermal resistance, several points limit its predictive capability. These limitations are mainly related to the fact that other important system parameters, such as soil volumetric heat capacity and grout thermal properties, must be considered as known inputs in the estimation procedure. The analysis presented here highlights that significant improvements have been achieved with the use of numerical tools. These tools have enabled the solution of the complete equation set describing the conductive-convection phenomena occurring in the system by avoiding the necessity to resort to approximations, which impair the accuracy of the modeling approach.

Under a comparative approach, it must be noted that analytical models, particularly the LSM and CSM, show several advantages: they are fast and easy to use and allow the estimation of the soil thermal conductivity and of the borehole thermal resistance by adopting the late regime temperature history. At the same time, this simple approach is not able to model the borehole geometry since the heat source is lumped in a line or a cylindrical source placed in the system axis.

On the other hand, the numerical modeling approaches, both to assist the TRT and to size the complete geothermal system, enable more complex phenomena to be taken into account: additional transport mechanisms other than conduction (for instance, groundwater advection within the soil), variable heat flow rate applied to the carrier fluid, variable mass flow rate of the fluid, temperature oscillations of the soil surface, and detailed geometrical configuration. At the same time, in their application, more awareness on the underlying physical phenomena and, consequently, more expertise in both the numerical computation heat transfer and fluid dynamics and parameter estimation procedures are required. Moreover, the computational cost increases significantly, especially if a three-dimensional modeling approach is adopted.

Additionally, it is noted that scientific efforts have been mainly devoted to achieving advances in these modeling approaches by providing transient solutions of the three-dimensional form of the governing equations by additionally accounting for complex geological settings, whereas less attention has been paid to the estimation procedure problem, which necessarily completes the TRT analysis. In particular, little attention has been given to the optimization of parameter estimation methods, which have only recently been applied to recover the thermal conductivities and the volumetric heat capacities of both the soil and of the filling material.

DOI: 10.1080/10789669.2011.610282


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Sara Rainieri, * Fabio Bozzoli, and Giorgio Pagliarini

Department of Industrial Engineering, University of Parma, Parco Area delle Scienze, 181/A 1-43124 Parma, Italy

* Corresponding author e-mail:

Received February 25, 2011; accepted July 4, 2011

Sara Rainieri, PhD, Member ASHRAE, is Associate Professor. Fabio Bozzoli, PhD, is Full Researcher. Giorgio Pagliarini Member ASHRAE, is Full Professor
Table 1. Most relevant literature contributions in which the inverse
problem approach is applied to the TRT.

Reference Approach

Mogensen (1983) LSM
Eskilson(1987) LSM
Gehlin and Hellstrom (2003b) LSM, CSM, 1D numerical
Kavanaugh and Rafferty (1997) CSM
Yavuzturk et al. (1999) 2D numerical horizontal slice
Shonder and Beck (1999) 1D numerical horizontal slice
Shonder and Beck (2000a) 1D numerical horizontal slice,
Shonder and Beck (2000b) 1D numerical horizontal slice,
Austin et al. (2000) 2D numerical horizontal slice
Witte et al. (2002) 2D numerical horizontal slice, LSM
Beier and Smith (2003) Laplace transform method, LSM
Roth et al. (2004) LSM
Sanner et al. (2005) LSM
Wagner and Clauser (2005) 2D numerical vertical slice
Signorelli et al. (2007) 3D numerical, LSM
Marcotte and Pasquier (2008) 3D numerical
Esen and Inalli (2009) LSM
Fujii et al. (2009) CSM
Sharqawy et al. (2009) LSM
Gustafsson and Westerlund (2010) 1D numerical horizontal slice
Bozzoli et al. (2011) 3D numerical
Raymond et al. (2011c) LSM, 3D numerical

Reference Unknown parameters

Mogensen (1983) [R.sub.b], [[lambda].sub.s]
Eskilson(1987) [R.sub.b], [[lambda].sub.s]
Gehlin and Hellstrom (2003b) [R.sub.b], [[lambda].sub.s]
Kavanaugh and Rafferty (1997) [[lambda].sub.s], [C.sub.s]
Yavuzturk et al. (1999) [[lambda].sub.s], [[lambda].sub.g]
Shonder and Beck (1999) [[lambda].sub.s], [[lambda].sub.g]
 volumetric heat capacity of a
 thin film simulating the fluid
 and pipe thermal inertia
Shonder and Beck (2000a) [R.sub.b], [[lambda].sub.s]
Shonder and Beck (2000b) [[lambda].sub.s], [[lambda].sub.g]
Austin et al. (2000) [[lambda].sub.s], [[lambda].sub.g]
Witte et al. (2002) [[lambda].sub.s], [[lambda].sub.g]
Beier and Smith (2003) [R.sub.b], [[lambda].sub.s]
Roth et al. (2004) [R.sub.b], [[lambda].sub.s]
Sanner et al. (2005) [R.sub.b], [[lambda].sub.s]
Wagner and Clauser (2005) [[lambda].sub.s], [C.sub.s]
Signorelli et al. (2007) [[lambda].sub.s]
Marcotte and Pasquier (2008) [R.sub.b], [[lambda].sub.s]
Esen and Inalli (2009) [R.sub.b], [[lambda].sub.s]
Fujii et al. (2009) [[lambda].sub.s] as a function of
Sharqawy et al. (2009) [[lambda].sub.s], [C.sub.s], soil
 thermal resistance
Gustafsson and Westerlund (2010) [R.sub.b], [[lambda].sub.s]
Bozzoli et al. (2011) [[lambda].sub.s], [C.sub.s],
 [[lambda].sub.g], [C.sub.g]
Raymond et al. (2011c) [[lambda].sub.s], [C.sub.s]
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Author:Rainieri, Sara; Bozzoli, Fabio; Pagliarini, Giorgio
Publication:HVAC & R Research
Geographic Code:1U2NY
Date:Nov 1, 2011
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