Geothermal Vertical Effects in Thermal Response Tests

We present analytical solutions for the temperature around the borehole heat exchanger which account for: (I) finite length line-source; (II) natural vertical gradient of the ground temperature; (III) arbitrary changes of the ground surface temperature. We find that the time-dependent part of the solution decays as a power of the time approaching the steady state limit. The long-time limit and asymptotic approach to this limit differ from that predicted by the infinite line-source model for very large times. We present series for the temperature in time scaled by the longitudinal and the transverse borehole lengths that also enables to determine the thermal conductivity of the ground, that is a critical parameter in designing geothermal borehole, and clarifies the estimation of the duration of the thermal response test. SECTION 1 – INTRODUCTION The Kelvin's infinite line source model is the most widely used one for the determination of the ground thermal conductivity (Mogenson 1983, Claesson et al 1987, Hellström 1991). For large values of time, finite size effects are to be taken into account; otherwise the ground temperature changes all the time. That is not the case for the finite line-source: the solution has been obtained and expressed as onedimensional integral (Claesson et al 1987, Carslowand et al 1971) given zero temperature at the boundary of semi-infinite medium. However, analytical asymptotic form of the solution is highly desirable to improve the estimation of the effective conductivity of the ground and the thermal resistivity of the borehole from data of the thermal response test (TRT). The results of such probes can be compared to the predictions based on the analytical formulas for the temperature as a function of time. Recently special attention is devoted to the estimation of temperature around the borehole by averaging over its longitude (Zeng et al 2002, Lamarche et al 2007). Typically, to this end, one applies the so-called "g" function introduced by Claesson and Elkinson (1987). The “g"function presents the thermal response factor of the borehole by the dimensionless temperature of the wall (Hellström 1991) and is used for modeling 3-D temperature distribution about the borehole. This "g"-function methodology is implemented in the simulation package TRNSYS, and in the software tools EED, GLHEPRO (Spitler 2000), which are commonly used for design of geothermal heat pump systems. In addition to the widely used numerical methods, analytical approach allows to check physical processes behind a model and analyze how the solution varies with the parameters of the designed system. Further refinement is desirable, because of the fact that numerical calculations show that the evaluation at the middle of the borehole overestimates its steady state temperature (Hellström 1991, Zeng et al 2002, Lamarche et al 2007). Although modeling borehole as a thin ellipsoid (Hellström 1991) and empirical formula for the steady state temperature as function of rb/H (based on the numerical results of the linesource theory) (Zeng et al 2002) mitigate this problem, still the mean integral temperature remains the best solution for engineering purposes (Zeng et al 2002, Lamarche et al 2007). Following this suggestion (Zeng et al 2002) a new expression for the "g"-function was introduced in order to increase accuracy and to decrease computational cost. The averaged exact solution for temperature was modified from the double to one-dimensional integral form (Lamarche et al 2007). Such "g" function, though available and not computationally expensive, is still clumsy to use. In this paper we pursue the following purposes: (I) Elaboration of the analytical formulas to determine in situ thermal conductivity and thermal resistance of a borehole by multi-variable parameter fitting; (II) Selecting an efficient model for the evaluation of the TRT data; justifying the usage of the mean temperature for self-consistent interpretation of the TRT data; (III) Obtaining approximations for both mean and middle temperature of the borehole; estimating the time of attainment of a steady state. This paper presents two main contributions. First, we introduce a more efficient version of the estimation of the thermal conductivity and the thermal resistance. This strategy solves the longstanding problem of that the steady-state temperature at the middle of the borehole is overestimated in the traditional approach. Second, we conduct an exhaustive comparison of the above methods of assessment the borehole wall temperature. For this purpose, we also report our results on the series in time for the temperature about the middle point of a borehole. The rest of the paper is organized as follows. Section 2 introduces the standard formulations for the finite line-source model, and presents the solutions obeying general boundary and initial conditions, highlights the limitation of infinite line-source model causing from infiniteness assumption. Section 3 first reviews the results of the classical infinite line-source theory, and then proposes a method to evaluate TRT data based on the integral mean of the borehole temperature for consistent account of the axial heat transfer effects. Section 4 summarizes our present results, compares them with the major features that emerge from evaluating borehole temperature at its middle. The findings are then analyzed theoretically and through illustrative examples. Finally, Section 5 concludes and gives some directions for further investigation. SECTION 2 LINE SOURCE THEORY The notation used throughout this paper is summarized in Table 1. A comparative study between numerical results and results of the infinite line-source (ILS) model showed that it is valid under some conditions (Signorelli et al 2007), which we specify below. Within this framework, commonly applied for the estimates of thermal response test data, the infinite medium is considered at given undisturbed ground temperature,T0. In this paper, we consider natural flow along the vertical z axis with a constant gradient geo zT ∇ in semi-infinite region, ground surface temperature of which,ψ (t), varies with time. We assume medium to be homogeneous and denote its volumetric heat capacity by C and its coefficient of thermal conductivity by λ, C λ α = represents the thermal diffusivity. The heat is released at a constant rate along the borehole heat exchanger (BHE), and is transferred by mechanism of thermal conductivity. Within the finite line-source (FLS) model the equation of heat diffusion is invariant under spatial rotation about z axis of the Notation Description H Height of the borehole heat exchanger, [m] rb Radius of the borehole heat exchanger, [m] f M Mass flow rate of heat carrier fluid, [kg/s] Rb Thermal resistance between fluid and borehole wall T Temperature of ground, [C] T0 Reference temperature (of undisturbed ground), [C] Tin Inlet temperature of BHE [ C] Tout Outlet temperature of BHE [ C] geo zT ∇ Geothermal gradient, [Km ] Qz Heat flux per unit length, [Wm ] C Volumetric heat capacity, [JmK] λ Thermal conductivity of ground, [WKm] C λ α = Thermal diffusivity of ground, [m s] b(f) Subscript associated with the borehole wall (fluid) S Subscript associated with the steadystate Table 1. Notation in Latin and Greek letters used in this paper. vertical BHE. The temperature of the ground, T, is defined by the heat conduction equation. )) ( ) ( )( ( ) , , ( ) , , ( H z z r Q t z r T t t z r C z − − + ∆ = ∂ ∂ ⊥ ⊥ ⊥ θ θ δ λ r r r at 0 , 0 ≥ ≥ z t (1) where coordinate vector ⊥ r r is orthogonal to z axis, Qz is the heat flux density per length unit of the BHE of radius b r and θ(z) is the step function. The initial condition, z T T z T t z T geo z ∇ + = = = 0 0 ) ( ) 0 , ( , reflects natural flow; the constant geo zT ∇ is known as the geothermal gradient. The boundary condition on the surface, ) ( ) , 0 ( t t z T ψ = = , captures the rather pronounced effect of the variation of the ambient air temperature with time (Sanner et al 2007) on the upper part of the BHE. We write temperature as a sum of the solution of the inhomogeneous Eq. (1), d v , and solutions of the homogeneous Eq. (1), s v v , 0 , which have to be determined; these satisfy the conditions indicated in Table 2. The solutions for the mixed problem with the prescribed temperature, ) (t ψ , on the surface of the semi-infinite medium with natural flow are presented in Table 3. Conditions at ground surface ( ) 0 , 0 , = = t z r vd ( ) ) ( , 0 , t t z r vs ψ = = ( ) 0 , 0 , 0 = = t z r v Initial conditions ( ) 0 0 , , = = t z r vd ( ) 0 0 , , = = t z r vs ( ) z T T t z r v geo z ∇ + = = 0 0 0 , ,