Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 11-13, 2013 SGP-TR-198 INFLUENCE OF CONVECTION ON PRODUCTION FROM BOREHOLE HEAT EXCHANGERS Carina Bringedal, Inga Berre, Jan Nordbotten Department of Mathematics University of Bergen 5020 Bergen, Norway e-mail: [email protected] ABSTRACT Convection cells in a porous medium form when the medium is subjected to sufficient heating from below (or equivalently, cooling from above) or when cooled or heated from the side. In the context of geothermal energy extraction from a sealed system, we are interested in how the convection cells transport heat when the porous medium is subjected to a cool, sealed borehole, also known as a borehole heat exchanger. Using pseudospectral methods together with domain decomposition we consider two scenarios; one system initialized with constant temperature and one system initialized with a vertical temperature gradient in the rock surrounding the borehole heat exchanger. We find the convection cells to have a positive effect on the heat extraction for a constant initial rock temperature, but a negative effect for some of the systems with an initial temperature gradient in the rock: Convection gives a negative effect when the borehole temperature is close to the rock temperature in the borehole, but gradually provides a positive effect when the borehole temperature gets colder and the convection stronger. INTRODUCTION Borehole heat exchangers (BHEs) are utilized for production of shallow and medium depth geothermal energy. The subsurface unit of a BHE consists of a sealed vertical pipe containing a production fluid that is circulated inside the pipe; cold fluid is pumped downwards and gets heated by the warmer ground, while the heated water is pumped back up giving a net energy profit. Many buildings have shallow boreholes producing local heating (and cooling) in combination with heat pumps, and deep boreholes are used for direct heating (Rybach and Hopkirk, 1995; Kohl et al., 2002). In the porous medium surrounding the borehole, several thermal processes can be present. Conduction will always transfer energy from regions with higher temperature towards regions with lower temperature. If the porous medium is saturated with fluid, typically water, that is allowed to flow in the porous medium, convection is also possible. As the density of water depends on temperature, buoyancy forces can generate groundwater flow without any outside force; this is called natural convection. Natural convection creates convection currents, which are circulation patterns for the saturating fluid. These currents will also transfer heat, but how and how much is dependent on flow properties of the fluid and the geometry of the porous medium. In this paper we focus on how induced convection currents affect the production from a BHE. When a BHE is producing heat, the borehole is filled with fluid having a temperature that is lower than the surrounding porous medium. This temperature difference will trigger convection currents in the surrounding porous medium as a horizontal temperature gradient always causes fluid motion (Vadasz et al, 1993). It is not obvious how the convection currents will contribute to the heat extraction from a BHE, as they may either contribute to the heat production by transporting more energy towards the borehole, or give a negative effect by transporting the heat energy away from the borehole. To isolate the effect of the natural convection, we assume the system to have no net groundwater flow. A forced advective flow would also affect the heat production, and has been investigated by e.g. Eskilson (1987), Chiasson et al. (2000) and Diao et al. (2004), but will not be considered here. MODEL FORMULATION We study an idealized geothermal system containing a single heat producing BHE. The borehole is sealed in the sense that there is no injection or production in the reservoir, which is typical for shallow and medium depth systems used mainly for heating and cooling applications. Our borehole is of a coaxial type where the water flows downwards in the outer tube and upwards in the inner tube. The coaxial borehole is used for both shallow and deep systems, as seen in Kohl et al. (2002) and Diersch et al. (2011). Other borehole structures, such as a U-type, could be applicable for the same uses, but is not studied here. The borehole is surrounded by a threelayered porous medium and produces heat only from the middle layer. The middle layer is permeable and saturated with water, while the top and bottom layer is assumed to be heat conducting, unsaturated rock or soil, acting as heat reservoirs for the saturated layer. For simplicity we assume the three porous layers to be homogeneous and isotropic. We consider two different models: In the first model the initial temperature variations in the surrounding porous medium are neglected. This is the case if the fluid inside the borehole is much colder than temperature in the reservoir, making the initial temperature variations in the reservoir less important. For example, if the initial ground temperature in the ground varies between 20C and 25C while the production fluid has an inlet temperature of 5C or lower, then this model is applicable. The other model is initialized with a vertical temperature gradient and represents a system where the initial temperature variations are too large, compared to the borehole temperature, to be neglected. If the initial ground temperature varies between 45C and 65C, while the inlet temperature in the borehole is 35C, which is the case studied by Kohl et al. (2002), the initial temperature variations in the porous medium must be taken into account. For the rest of the article these two models will be known as the constant temperature (CT) model and the temperature gradient (TG) model, respectively. Our domain is shaped as an annular cylinder and is sketched in Figure 1. The three porous layers 1, 2 and 3 have heights , and , respectively. We only model the downward flow inside the BHE, which is contained between and . Assuming the casing between the inner and outer tube to be insulated, the flow in the inner tube does not affect the heat production. Also, the BHE is assumed to only extract heat from layer 2. Figure 1: Cross-section of domain. Governing equations To describe the fluid flow in the saturated layer, we assume Darcy’s law, ( ) (1) where v is the fluid velocity, is the permeability of the porous medium, and are the viscosity and density of the groundwater, respectively, is the gravitational acceleration, and k is the vertical unit vector pointing upwards. The density is given by [ ( )], (2) where at some reference temperature , and is the thermal expansion coefficient. Note that the density of water would normally also be dependent on pressure, but table 4 in Fine and Millero (1973), reveals that for temperature and pressure domains relevant for geothermal energy, density variations due to temperature is the most dominating. Even though density variations are present, they are so small that we apply the Boussinesq approximation, which states that density differences in the fluid can be neglected unless they occur together in terms multiplied with the gravity acceleration . Hence, we can apply the mass conservation equation for an incompressible fluid; (3) We further assume energy conservation for both fluid and solid in layer 2; that is ( ) ( ) (4) In the above equation, subscript refers to the fluid and to the medium. Furthermore, ( ) is the overall heat capacity per unit volume where is the specific heat, and is the overall thermal conductivity of the fluid and solid combined. When we use the overall heat capacity and the overall thermal conductivity, we are using porosity-weighted averages of the heat capacities and thermal conductivities of the fluid and solid. Finally, is the temperature of fluid and solid. No equations are needed to describe the fluid flow in the lower and upper layer; hence, only ( ) (5) has to be solved here. The subscript refers to the solid. The heat conductivity of the solid is in this equation denoted to emphasize that it could be different from the heat conductivity of the solid in layer 2. The temperature inside the outer tube of the borehole is modeled using ( ) ( ) (6) The fluid velocity inside the borehole is assumed to be equal to the injection velocity. To estimate the effect of convection on the heat production, we calculate the heat flux into the borehole using (7) where is the heat flux, is an outward unit normal for the borehole, and is the surface element on the borehole. The integral is to be taken over the whole borehole facing layer 2. Nondimensional equations and the Rayleigh number To nondimensionalize the equations, we use the following coordinate transform: (8) ( ) ( ) is the thermal diffusivity, and Here, ( ) ( ) is the ratio of the volumetric heat capacities of medium and fluid. The two temperatures and are reference temperatures and should represent a typical temperature difference in the system. In the CT model, and will be the initial temperatures in the porous medium and in the borehole, respectively. In the TG model, and are the initial temperatures at the bottom and top of the saturated layer. The superscript denotes that the variable has no dimension. Substituting the above nondimensional variables into the governing equations, introduce the dimensionless Rayleigh number ( ) (9) where is the kinematic viscosity of the saturating fluid. The Rayleigh number works as a measure of the strength of the convection; a Rayleigh number of zero corresponds to the saturated layer being impermeable and provides no convection. A Rayleigh number larger than zero would always provide convection due to the horizontal temperature gradient (Vadasz et al., 1993). If there is no horizontal temperature gradient, convection currents can only develop if the Rayleigh number is larger than some critical Rayleigh number, see (Bringedal et al., 2011) for a further discussion. Substituting the dimensionless variables into our model equations yields a new system of equations. Darcy’s law (1) is transformed into (10) the mass conservation equation (3) becomes (11) the energy conservation equation for layer 2, equation (4), becomes (12) while for the lower and upper layer, equation (5), is (13) The energy conservation equation for the borehole (6) is now (14) Initial and boundary conditions To solve the equations, initial and boundary equations are required for temperature, and also boundary equations for velocity. The CT model is initialized with the constant temperature in the porous domain and with in the borehole. Hence, is also the injection temperature in the borehole at all times. All outer boundaries around the porous medium is held perfectly heat conducting at , while the top of the borehole is kept at and otherwise insulated. In the connection between borehole and porous medium, and also in the connection between the layers, continuity in temperature and heat flux is required. The TG model is in the porous medium initialized with a linear temperature distribution corresponding to at the bottom of layer 2 and at the top of layer 2. The borehole is initialized with an injection temperature . The outer boundaries belonging to the porous domain and the top boundary of the borehole are all kept perfectly heat conducting corresponding to their initial condition. The left hand side and bottom of the borehole is kept insulated, and we require continuity in temperature and in heat flux at the internal boundaries as in the CT model. For both CT and TG model, we require boundaries of layer 2 in the porous domain to satisfy a no-slip condition for the velocity. In the borehole, the velocity is given by ( ) and satisfies no-slip condition at the vertical boundaries. NUMERICAL SOLUTION APPROACH An unsteady 3D-solver that approximates the solution of the nonlinear equations (10)-(14) has been written using pseudospectral methods in space and MATLAB’s built-in package ODE15s in time. Pseudospectral methods are higher-order numerical methods known for their good convergence properties. We have chosen this method to obtain high resolution of the temperature distribution near and inside the borehole, which is essential for investigating small changes in the heat flux into the borehole. In earlier work we have successfully applied these methods to investigate onset and stability of convection currents (Bringedal et al., 2011). Here we give a short review of the pseudospectral methods and refer to Boyd (2001) for a more thorough introduction. Pseudospectral methods is a collocation methods that search for the function values in the collocation points, and is in that sense related to finite difference methods. However, pseudospectral methods obtain very high accuracy by placing the collocation points in a specific manner, and the optimal choice of grid points depends on the geometry of the domain. As we use cylindrical coordinates, we apply the Chebyshev nodes in the radial and vertical direction and the Fourier nodes in the azimuthal direction. These choices will provide the fastest possible convergence, as explained by Boyd (2001). In the resulting matrix equation when discretizing with pseudospectral methods, each line represents an equation for the function value in one of the collocation points. Applying boundary equations is then only a matter of localizing collocation points at the boundaries and insert a discrete version of the boundary condition. As we have internal boundaries both between the layers in the porous medium and between the borehole and the porous medium, we apply domain decomposition to ensure that we have collocation nodes also at the boundaries. We divide in four subdomains equal to the four rectangles sketched in Figure 1; one for each layer and one for the borehole. The decomposing also enables us to use a finer grid in any of the layers or in the borehole if necessary. The subdomains are stitched together using the continuity requirements described within initial and boundary conditions. RESULTS AND DISCUSSION The governing equations (10)-(14) are solved by timestepping the energy equations (12)-(14) and updating the velocity field in the porous medium using Darcy’s law (10) and the mass conservation equation (11) in each time step. The production period for a BHE depends on geological conditions and usage, but is expected to be less than 50 years for deeper boreholes, while shallow boreholes used for direct heating are normally closed or applied for cooling purposes after each winter season. In dimensionless time, both these time spans correspond to . All following results are given at . We need to apply values for the parameters listed in equations (10)-(14). For , , and we apply values realistic for groundwater and ground, while for the Rayleigh number we use a range of representative values. In the papers by Diersch et al., (2011), Lazzari et al. (2010) and Kohl et al. (2002) values for injection velocity for both shallow and deep systems can be found. Translated to dimensionless injection temperature, this corresponds to . The same three papers provide values for the borehole diameter, which in dimensionless variables results in . The values of , and are chosen to be so large such that the outer boundaries will not have a significant effect on the results in the relevant time period. For the TG model, a range of values for the injection temperature is needed. For a complete list of values of parameters, see Table 1. Table 1: Values or ranges of values used for parameters in simulations. Parameter ( ) (only for TG model) Value/Range ( ) The CT model In the CT model the convection currents, when present, would always distribute such that hot groundwater is transported towards the upper half of the borehole, then down along the borehole while heat diffuses into the borehole and cools the groundwater, and then transported away from the borehole at the bottom of layer 2. See Figure 2 for temperature distribution near the borehole for a convective and a non-convective case. Figure 3 the obtained heat fluxes Rayleigh numbers is showed. for various Figure 3: Nondimensional heat fluxes into the borehole for various Rayleigh numbers. From Figure 3 we see that the heat flux increases with increasing Rayleigh number, but the difference is not large. We have also performed simulations where we have tried other values for , , , and corresponding to other scenarios, but the results in heat flux where qualitatively the same; the heat fluxes still increase with increasing Rayleigh number. The TG model The convection currents distribute in the same manner in the TG model; groundwater flow towards the borehole in the upper half of layer 2 and away from the inner cylinder in the lower half. See Figure 4 for temperature distribution near the borehole for a convective and non-convective case. Figure 2: Temperature distribution for Ra = 0 (top) and Ra = 100 (bottom). The colored lines are isotherms where red is colder and yellow warmer. The arrows indicate groundwater velocity. The simulations shown in Figure 2 were made using , , , and ( ) , corresponding to a shallow borehole. From Figure 2 it is clear that convection currents will provide slightly larger heat production in the upper half of layer 2 as warm fluid is transported towards the borehole here. In the lower half of layer 2 the heat flux becomes slightly smaller, but altogether the convection currents gives larger heat production. In Figure 5: Nondimensional heat fluxes into the borehole for various Rayleigh numbers when . From Figure 5 we see that the heat flux decreases significantly with increasing Rayleigh number. We have also performed simulations where we have tried other values for , , , and corresponding to other scenarios, but the results in heat flux where qualitatively the same; the heat fluxes still decrease with increasing Rayleigh number. However, lowering revealed a significant effect. With a lower value of , the initial temperature variations in the ground water is less important. Convection will still transport the colder upper lying fluid towards the borehole, but even the coldest groundwater will be significantly warmer than the borehole. Especially for larger Rayleigh numbers, the dominating effect when using a low is that the convection currents provide more heat transport and give a larger heat flux into the borehole. Heat fluxes for various Rayleigh numbers and is given in Figure 6. Figure 4: Temperature distribution for Ra = 0 (top) and Ra = 100 (bottom). The simulations shown in Figure 4 were made using the same parameters as for Figure 2, and with . This value for corresponds to the injection temperature being very close to the coldest ground temperature from where heat is extracted and is an extreme case, but is not uncommon for longterm heat extraction. From Figure 4 we observe that the convection currents now give a negative effect; even though there is more heat transport in the system, it is the colder, upper lying fluid that is transported towards the borehole. In Figure 5 the obtained heat fluxes for various Rayleigh numbers are showed. Figure 6: Nondimensional heat fluxes into the borehole for various Rayleigh numbers when (white), (light grey), (dark grey) and (black). For borehole injection temperatures lower than 0, we see two trends in Figure 6: First of all we get in general larger heat fluxes for all Rayleigh numbers; this is due to there being a larger temperature difference between borehole and ground present. However, if we compare heat fluxes for the same values of we see that for larger Rayleigh numbers, the convection can give a positive effect on the heat production. For gradually colder , this occurs for even smaller Rayleigh numbers. This is as expected since the TG model with a very low value of should act as the CT model; when becomes significantly lower than 0 the initial temperature variations in the ground becomes less important and we could in stead use the CT model to describe the system. Simulations with between ( ) and ( ) gave results showing the same trend as observed for the CT model. CONCLUSIONS Using pseudospectral discretization together with domain decomposition, we have made a solver for modeling the heat transfer processes in a layered porous medium when a sealed borehole is extracting heat. Our high-order numerical simulations show how the heat transfer into a borehole heat exchanger is affected by the presence of natural convection currents. For a system using a constant temperature distribution as initial condition in the porous medium, convection currents will give a small positive effect to the heat extraction as the currents retrieve some extra heat towards the borehole. Stronger convection will always result in a larger heat flux into the borehole. For a system having a vertical temperature gradient as initial condition in the porous medium, the effect of convection depends on the injection temperature in the borehole. When the injection temperature is close to the coldest temperature in the saturated porous medium, the convection will give a negative effect on the heat production as the cold groundwater is transported towards the borehole giving a very small heat flux. For stronger convection, the heat flux becomes even smaller. For colder injection temperatures in the borehole, the situation gradually changes as the initial temperature variations in the subsurface become less significant and the system acts more like the model initialized with a constant temperature in the porous medium. For borehole heat exchangers our results affect the choice of injection temperature in the borehole. Using an injection temperature close to the groundwater temperature is typical for long-term BHEs as this results in the ground not cooling down so quickly, but we see here that this results in less heat production than expected if the ground is permeable and saturated with water so that convection currents can evolve. If using an injection temperature much lower than the ground temperature, such that in the model initialized with a temperature gradient in the porous medium, or such that the porous medium can be considered having a constant temperature before production from the BHE starts, the convection currents will give a positive effect on the heat extraction as more heat is transported towards the borehole. If the borehole is used for cooling purposes, such that the borehole temperature is warmer than the ground, the results are qualitatively the same; convection will provide a negative effect if the injection temperature is close to the initial ground temperature, and a positive effect is the injection temperature is significantly higher than the ground temperature. REFERENCES J. P. Boyd (2001), “Chebyshev and Fourier spectral methods”, Dover Pubns, New York. C. Bringedal, I. Berre, J. M. Nordbotten and D. A. S. Rees (2011), “Linear and nonlinear convection in porous media between coaxial cylinders”, Physics of Fluids 23 (9) 094109. A. D. Chiasson, S. J. Rees and J. D. Spitler (2000), “A preliminary assessment of the effects of groundwater flow on closed-loop ground source heat pump systems”, Tech. rep., Oklahoma State Univ., Stillwater, OK (US) . N. Diao, Q. Li and Z. Fang (2004), “Heat transfer in ground heat exchangers with groundwater advection”, International journal of thermal sciences 43 (12) 1203–1211. H.-J. G. Diersch, D. Bauer, W. Heidemann, W. Rühaak and P. Schätzl (2011), “Finite element modeling of borehole heat exchanger systems Part 2. Numerical simulation”, Computers and Geosciences 37 (1136-1147. P. Eskilson (1987), “Thermal analysis of heat extraction boreholes”, Department of Mathematical Physics, University of Lund,. R. A. Fine and F. J. Millero (1973), “Compressibility of water as a function of temperature and pressure”, The Journal of Chemical Physics 59 5529. T. Kohl, R. Brenni and W. Eugster (2002), “System performance of a deep borehole heat exchanger”, Geothermics 31 687-708. S. Lazzari, A. Priarone and E. Zanchini (2010), “Long-term performance of BHE (borehole heat exchanger) fields with negligible groundwater movement”, Energy 35 49664974. L. Rybach and R. Hopkirk (1995), “Shallow and deep borehole heat exchangers – achievements and prospects” Proceedings of World Geothermal Congress 1995, Florence, Italy, Vol. 3,, pp. 2133–2138 P. Vadasz, C. Braester and J. Bear (1993), “The effect of perfectly conducting side walls on natural convection in porous media”, International journal of heat and mass transfer 36 (5) 1159–1170.