Wellbore Heat Transfer in CO 2 -Based Geothermal Systems

Transcription

1 GRC Transactions, Vol. 36, Wellbore Heat Transfer in CO -Based Geothermal Systems Jimmy B. Randolph, Benjamin Adams, Thomas H. Kuehn, and Martin O. Saar Department of Earth Sciences, University of Minnesota, Minneapolis MN Department of Mechanical Engineering, University of Minnesota, Minneapolis MN Keywords CO Plume Geothermal, CPG, geothermal, carbon dioxide sequestration, numerical simulation, heat transfer, EGS Abstract Geothermal systems utilizing carbon dioxide as the subsurface heat exchange fluid in naturally porous, permeable geologic formations have been shown to provide improved geothermal heat energy extraction, even at low resource temperatures, compared to conventional hydrothermal and enhanced geothermal systems (EGS). Such systems, termed CO Plume Geothermal (CPG), have the potential to permit expansion of geothermal energy use while supporting rapid implementation. While most previous analyses have focused on heat transfer in the reservoir and surface components of CO -based geothermal operations, here we examine wellbore heat transfer. In particular, we explore the hypothesis that wellbore flow can be assumed to be adiabatic for the majority of a CPG facility s lifespan. Introduction Part of the effort to limit CO emissions to the atmosphere requires that existing and emerging energy systems shift away from using fossil fuels as primary energy inputs and towards renewable sources that may be used to generate electricity. Renewable sources of primary energy, including geothermal heat, offer the potential to produce electricity with little or no operational CO emissions. The transition towards these renewable energy sources will be slow, in part because of embedded infrastructure that naturally turns over on the order of half a century, and in part because the high energy density of hydrocarbons and well-developed hydrocarbon power systems result in produced electricity that is inexpensive relative to other options. Given that CO emissions are increasing and unlikely to be abated by sudden and dramatic changes in how societies produce, convert, and use energy, applications that can use CO -- particularly to produce electricity -- may enable transitions to renewables that are not disruptive to economies and the climate. Numerous previous studies have discussed using supercritical CO as the subsurface working fluid for geothermal energy capture (e.g., [Brown, ], [Pruess, 6; 8], [Atrens et al., 9], [Randolph and Saar, ; ]). Numerical analyses suggest CO transfers heat more efficiently than water, particularly in naturally permeable and porous geologic formations, i.e., in CO -Plume Geothermal (CPG) systems [Randolph and Saar, ; ], as a result of its relatively high mobility (inverse kinematic viscosity) and compressibility in deep geologic formations, the latter contributing to a strong thermosyphon [Atrens et al., 9, Adams et al., in preparation]. Therefore, CO -based geothermal operations may permit use of lower temperature and lower permeability geologic formations than those currently deemed economically viable. Direct power system Confining unit (e.g., shale) Enhanced hydrocarbon recovery (EOR) Confining unit Saline aquifer Crystalline basement Heat exchanger CO capture District heating CO source (fossil fuel power plant, biofuel plant, etc) CO plume geothermal Binary power system CO plume geothermal (CPG) EGS Cooling tower Expansion device/generator Figure. Several potential implementations of CO -based geothermal systems (modified after Randolph and Saar []). 549

2 CPG systems involve injecting CO -- produced by hydrocarbon power systems and/or other industries -- into deep, naturally porous and permeable geologic formations where the CO displaces native reservoir fluid and is heated by the natural in-situ heat and the background geothermal heat flux. A portion of the heated CO is piped to the surface, providing energy for electricity production or direct heat utilization, before being returned to the subsurface. Enhanced geothermal systems (EGS) using CO as the subsurface heat transfer medium are designed very similarly to water-based EGS but, like CPG, achieve many of the advantages of CO over water (e.g., [Pruess, 6]). With either system, the injected CO is ultimately geologically stored, though CPG provides far larger CO storage capacities and higher energy electricity production efficiency [Randolph and Saar, ]. Previous CPG analyses [Randolph and Saar, ; ], as well as many CO -based EGS studies (e.g., [Atrens et al., 9], [Pruess, 6; 8]), have focused on subsurface heat transfer or surface components of the system while assuming minimal or no heat loss or gain in the production or injection wells, respectively, and no wellbore frictional losses. To ensure the accuracy of future CPG studies, wellbore flow and heat transfer assumptions -- in particular, that of adiabatic expansion/compression -- are examined in detail in the present investigation. In the wells, heat is transferred with surrounding rock, and frictional effects between the flowing fluid and the pipe wall result in pressure loss. Back-of-the envelope calculations reveal Reynolds numbers on the order of 7 for CO pipeflow, suggesting large advective heat transfer coefficients. However, this effect may be offset by the low effective thermal conductivity of the subsurface solid-fluid complex. Moreover, the low dynamic viscosity of supercritical CO, together with the relative smoothness of the nickel-steel pipe generally required for CO wells, suggest frictional losses are low. These opposing factors indicate further wellbore analysis is warranted and, thus, is presented below. Heat Transfer Model and Theory A simple, quasi-one-dimensional transient heat transfer model is generated, as shown in Figure, representing the buoyant ascent of heated CO from the reservoir to the surface. We assume that convective heat transfer outside of the wellbore is negligible. While future studies may allow consideration of convective flow, neglecting it is considered reasonable because reservoir formations chosen for CPG installations will, by necessity, be overlain by large extents of low-permeability Figure. One-dimensional heat transfer model schematic (not to scale). caprock -- ensuring injected CO is trapped in the reservoir -- with minimal potential for fluid flow and, hence, convective heat exchange. We further assume that temperature gradients in the vertical direction are much less than in the radial direction. Thus, vertical heat transfer is assumed negligible compared to radial transfer, permitting the model to be onedimensional in the radial direction. The model contains 6 radial grid blocks with radii of.5m, m, m, m, 4m, and m; and grid blocks in the axial direction at even increments of 5m. Finer radial and vertical grid refinements were tested but found to be unnecessary for the purpose of this investigation. We develop the model using Engineering Equation Solver (EES) for its simultaneous equation solving ability in a single axial slice and efficient retrieval of thermodynamic property data, where the latter is based on Span and Wagner [996]. EES is linked iteratively with MATLAB on a macro-level to process and store axial and time increment results. Heat transfer and frictional losses should be more significant in CPG production wells, as opposed to injection wells, because of higher fluid temperature and flow velocity for relatively low density produced CO. Thus, we focus here on production well behavior. The CO fluid conditions at the bottomhole well inlet are fixed at 5 MPa, C, and 8 kg/s, consistent with the base case of previous investigations [Randolph and Saar, ; ]. Heat transfer is found for a buoyant CO rise of.5 km, while the surrounding rock temperature is set initially to a linear geothermal gradient of 35 C/km and a mean annual surface temperature of.5 C. The radial temperature boundary condition at grid 6, m from the well, is set equal to this geothermal gradient. The pipe wall is neglected in the heat transfer Table. Nomenclature. h Specific enthalpy [kj/kg] = u + pυ q Heat transfer rate [kw] T res Reservoir temperature [ C] T amb Surface ambient temperature [ C] T Temperature [ C] P Pressure [MPa] h Specific enthalpy [kj/kg] z Depth [m] r Radial distance from well center axis [m] Δz Grid block thickness [m] D Well diameter [m] m Working fluid mass flowrate [kg/s] V Working fluid velocity [m/s] ρ Working fluid density [kg/m] z Working fluid elevation [m] µ Working fluid viscosity [kg/(m s)] k Working fluid thermal conductivity [W/(m K)] C p Working fluid specific heat capacity [kj/(kg K)] C rock Rock specific heat capacity [kj/(kg K)] ρ rock Rock density [kg/m 3 ] k rock Rock thermal conductivity [W/(m K)] V rock Volume of rock in a grid element [m 3 ] A Heat transfer area [m ] f Darcy friction factor [dim] Re Reynolds number [dim] Nu Nusselt number [dim] Pr Prandtl number [dim] h x Convective heat transfer coefficient [W/(m K)] g Gravitational acceleration [m/s ] 55

3 analysis, as its high thermal conductivity offers little resistance to heat transfer compared to the rock. The rock is assumed to have % porosity, and conservatively high thermal conductivity and heat capacity are assumed for the effective, combined water-rock system. For a summary of the nomenclature used in the following analysis, see Table, and for a full set of model parameters, see Table (Appendix). Three unique energy balance cases are applied across each axial slice: a fluid convection boundary condition case (for the rock element bounding the wellbore), a center-grid conduction-only case, and a prescribed temperature outer boundary condition. For the convective boundary case, consider the annular volume segment i, with inner radius (the wellbore) r i,inner, outer radius r i,outer, the radius of the next segment r i+,outer, and having thickness Δz. An energy balance across the element yields q in,convection q out,conduction = q storage, where the general forms of each are: q convection = h x A surface ΔT q conduction = k rock A cross sectional dt dr q storage = ρ V rock C rock dt. dt Here, A surface is contact area between fluid and rock, which permits convective heat transfer, and A corss-sectional is the contact area for conduction (for cylindrical coordinates, the latter is the area at the log mean radius). For cylindrical coordinates, the discretized conductive heat flow equation can be written as q conduction = π k rock Δz ΔT ln r i,outer r i,inner Combining these produces the energy balance for this element h x ( π r i,inner Δz) T favg,t T iavg,t ( ) π k rock Δz T i,t T i+,t ln r i+,outer + r i,outter r i,outer + r i,inner ρ rock π r i,outer ( ) = ( r i,inner ) Δz C rock T i,t+δt T i,t Δt Here, h x is the convective heat transfer coefficient, T favg,t is the average temperature of the fluid across the segment, T i,t is the temperature of the element at time, t, T i+,t is the temperature of the next element at time, t, T i,t+δt is the temperature of the element at the next timestep, T iavg,t is the average temperature of element i of timesteps t and t+δt, and Δt is the discrete timestep at which this numerical integration will be performed. We find that average temperatures between discretizations most accurately reflect the actual heat transfer that is taking place. For this analysis, Δt is. days for the initial 5 days, 36.5 days for the remainder of the first years, and 46 days for the remaining 3 years. For the two remaining conduction-only cases, q = q in, conduction qout, conduction ( ) π k rock Δz T i,t T i,t r ln i,outer + r i,inner r i,inner r i,inner ( ) π k rock Δz T i,t T i+,t ln r i+,outer + r i,outter r i,outer + r i,inner ρ rock π r i,outer = ( r i,inner ) Δz C rock storage T i,t+δt T i,t Δt In the case of the last element with the prescribed temperature boundary condition, T i+,t is set to be the unchanging surrounding rock temperature at depth. r i+,outer is also assumed such that the difference between r i+,outer and r i,outer is equal to the difference between r i,outer and r i,inner. The convective heat transfer coefficient, h x, is found using the Nusselt number, Nu, where D is the well diameter and k is the fluid thermal conductivity: h x = Nu k D. The Nusselt number is related to the Reynolds number, Re, and Prandtl number, Pr, using the following relationship for turbulent pipe flow [Dittus and Boelter, 93], by Nu =.3 Re.8 Pr.4. The Reynolds and Prandtl numbers are, respectively, Re = ρ V D µ and Pr = C p µ, k where the properties density, ρ, dynamic viscosity, µ, specific heat (at constant pressure), C p, and thermal conductivity, k, are evaluated -- using EES -- for the fluid and assumed constant within each element. Patched-Bernoulli is used to account for the hydraulic head change and frictional losses as the fluid rises, P + ρ V + ρ gz = P + ρ V + ρ gz + f Δz ρ V D, where subscripts and indicate consecutive elements within the wellbore, and the final term is the Darcy-Weisbach equation, whose friction factor, f, is determined from a moody chart. The mean fluid velocity in each element changes with fluid density changes and is evaluated using the continuity equation, i.e., ensuring conservation of mass. Similarly, to account for the energy decrease as fluid rises, an energy balance is performed between consecutively wellbore elements, 55

4 Model Results m h + V + gz =. m h + V + gz + q convection, The heat loss, temperature, and pressure of CO flowing in the production well are shown in Figure 3. During the five days following the onset of production, we can define an initial transient heating phase of well operation. Throughout this initial transient period, the first rock element bounding the fluid heats to the fluid temperature, a process that occurs quickly because of the large advective heat transfer coefficient of the fluid coupled with the large temperature difference between the fluid and the rock. After this initial transient period, the fluid production pressure is essentially at, and the production temperature is within C, of the adiabatic limiting case. Heat transfer continues during the duration of the well operation, but at a much lower rate, as all surrounding rock elements slowly warm. For comparison, Figure 4 shows heat loss together with production temperature and pressure for flowing water, using the same model grid and system conditions. Notice that, as with CO, most transient production behavior occurs within the first five days. However, water loses approximately twice as much heat to the surrounding rock as does CO, a consequence of its higher heat capacity together with its greater temperature difference between fluid and rock at shallow depths. The latter is a result of minimal cooling with decreasing pressure as water rises, compared to over 4 C of cooling (though far less pressure loss) in the rising CO. See Figures 5 and 7 for CO and water temperature profiles, respectively. The analysis in Figure 3 shows a maximum.6 MW of heat loss after the initial transient period, with wellbore heat losses at less than 5% for most of the well s operational life. For Fluid Heat Loss [MW] CO Production States and Heat Loss in Well with Time No Heat Exchange T = 58 C No Heat Exchange P =.8 MPa month.. Time [days] year Fluid Heat Loss Production Temperature Production Pressure years Production Temperature [C] or Production Pressure [MPa] Figure 3. CO production temperature and pressure, as well as fluid heat loss to surrounding rock, as a function of time. Notice that after the initial approximately five days of significant transient behavior, produced fluid conditions approach the limiting case of no heat transfer with surrounding rock. These results validate the assumption of adiabatic wellbore flow for the majority of system operation. Fluid Heat Loss [MW] H O Production States and Heat Loss in Well with Time No Heat Exchange T = 98.4 C No Heat Exchange P =. MPa month. -. Time [days] comparison, a direct CO power system will extract 9 MW of thermal energy from the ground, assuming the given reservoir temperature and depth with a 8 kg/s flowrate [Adams et al., in preparation]. Figure 3 also shows that while wellbore heat loss affects the production temperature, production pressure remains nearly unaffected by the heat transfer. This pressure result indicates that a direct CO power system -- where produced CO is passed directly through a turbine rather than exchanging heat to a secondary working fluid -- which relies on pressure rather than thermal energy, would be nearly unaffected by such heat loss. Figure 5 shows the temperature profile of CO as it rises from a depth of.5 km at C, and Figure 6 provides the corresponding heat transfer coefficient and related fluid properties. Produced CO is within C of the adiabatic limiting case within days of production initiation. The adiabatic limiting case in Figure 5 includes frictional losses in the wellbore, whereas the isentropic case includes no irreversibilities (i.e., no enthalpy change). The isentropic curve clearly reveals the Joule-Thomson behavior of rising CO -- CO expands and cools with decreasing pressure as it approaches the surface -- and in comparison to the curves that year Fluid Heat Loss Production Temperature Production Pressure years Production Temperature [C] or Production Pressure [MPa] Figure 4. H O production temperature and pressure, as well as fluid heat loss to surrounding rock, as a function of time. Depth [m] days CO Temperature at Well Depths with Time.5 days. day. days 3. days. days No Heat Transfer, i.e. Adiabatic Isentropic (Upper Limit) CO Temperature [C] Figure 5. CO production well temperature profiles at several times, with days being the profile at the onset of fluid production. 55

5 Depth [m] Changes in Heat Transfer of Rising Fluid k (.6 W/m-C Initial) Viscosity (4.6e-5 Pa-s Initial) Density (578 kg/m^3 Initial) hx (47 W/m^-C Initial) Velocity (.8 m/s Initial) Re (8.8e6 Initial) Nu ( Initial) Pr (.74 initial) % Change Figure 6. CO fluid parameter change as a function of depth, with respect to values at the onset of production. Legend entries from top to bottom correspond to lines from left to right. account for real fluid expansion to the surface, we see that CO production well behavior is near-isentropic within a few days. As previously noted, the effects of CO cooling on electricity generation efficiency are offset by high pressure at the production wellhead. Referring to Figure 6, the initial convective heat transfer coefficient is found to be very large: 47 W/m -C. For comparison, the heat transfer coefficient of a typical home is 3 W/m -C. This accounts for the very rapid initial transfer to the surrounding rock. However, the limiting value for heat transfer away from the fluid in the well is the temperature difference between fluid and surrounding rock, not the convective heat transfer coefficient. As such, heat loss is unaffected by the changes in convective heat transfer coefficient, shown in Figure 6, which is also partially tempered by inversely proportional changes in density and thermal conductivity. For comparison, Figure 7 shows water temperature profiles at several times following the onset of fluid production. Unlike CO, once the adiabatic limit is reached, water temperature as a function of depth is essentially constant, with only a small temperature drop as a result of frictional effects. Depth [m] H O Temperature at Well Depths with time days.5 days. day. days 3. days 5. days No Heat Transfer (Upper Limit) H O Temperature [ o C] Figure 7. H O production well temperature profiles at several times, with days being the profile at the onset of fluid production. Discussion This analysis demonstrates that heat transfer away from the production wellbore in a CO Plume Geothermal system is minimal within a few days of the onset of production. Under the assumptions of the present study, wellbore flow nears adiabatic conditions after approximately five days of system operation. Thus, new CPG studies can generally assume adiabatic well flow. To enhance the current study, future work will allow for advection in the rock surrounding the wellbore. Should a well pass through a zone of high permeability, groundwater flow may enhance heat transfer across the casing, necessitating careful well completion. Future work will also examine injection well behavior. In particular, it is of interest to consider injection of sub-zero degrees Celsius CO. Permitting such low heat rejection temperatures can greatly increase geothermal power plant operating efficiency, a unique feature of CO -based geothermal systems, however such wellbores must be properly insulated to prevent freezing of surrounding groundwater. By ensuring careful well design, the favorable thermodynamic properties of CO can be fully utilized to enhance the viability of CPG technology and expand the use of geothermal energy. Acknowledgements Research support was provided by the Initiative for Renewable Energy and the Environment (IREE), a signature program of the Institute on the Environment (IonE) at the University of Minnesota (UMN), and by the US Department of Energy (DOE) Geothermal Technologies Program under Grant Number DE-EE764. M. O. Saar is Chief Scientific Officer of Heat Mining Company (HMC) LLC and J. B. Randolph is a Scientific Advisor to HMC LLC. Any opinions, findings, conclusions, or recommendations in this material are those of the authors and do not necessarily reflect the views of the DOE, IREE, IonE, UMN, or HMC LLC. Patents regarding the CPG technology have been filed by UMN and licensed exclusively and worldwide to HMC LLC. References Adams, B., J.B. Randolph, J.M. Bielicki, T.H. Kuehn and M.O. Saar, in preparation. A self-developing thermosyphon for carbon dioxide plume geothermal energy. Atrens, A.D., H. Gurgenci, and V. Rudolph, 9. CO thermosiphon for competitive geothermal power generation. Energy & Fuels, v.3, pp Brown, D.,. A hot dry rock geothermal energy concept utilizing supercritical CO instead of water. Proceedings of the Twenty-Fifth Workshop on Geothermal Reservoir Engineering, pp.33 38, Stanford University, Stanford, CA. Dittus, F. and L. Boelter, 93. Heat Transfer in Automobile Radiators of the Tubular Type. University of California Publications in Engineering, v., pp Moody, L., 944. Friction Factors for Pipe Flow. Transactions of the ASME, v.66, pp Pruess, K., 6. Enhanced geothermal systems (EGS) using CO as working fluid a novel approach for generating renewable energy with simultaneous sequestration of carbon. Geothermics, v.35, pp

6 Pruess, K., 8. On production behavior of enhanced geothermal systems with CO as working fluid. Energy Conversion and Management, v.49, pp Randolph, J.B. and M.O. Saar,. Coupling geothermal energy capture with carbon dioxide sequestration in naturally permeable, porous geologic formations: A comparison with enhanced geothermal systems. Geothermal Research Council Transactions, v.34, pp Randolph, J.B., and M.O. Saar,. Combining geothermal energy capture with geologic carbon dioxide sequestration. Geophysical Research Letters, v.38, L4, doi:.9/gl4765. Span, R. and W. Wagner, 996. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to K at Pressures up to 8 MPa. Journal of Physical Chemistry, Ref. Data, v. 5, n.6. Appendix Table. Heat transfer model parameters. Reservoir Temp, T res C Ambient Temp, T amb.5 C Thermal Gradient 35 C/km Rock Porosity. Rock Thermal Conductivity, k rock 3 W/(m C) Rock Spec Heat, C rock.9 kj/(kg C) Rock Density, ρ rock 3 kg/m 3 Well Depth.5 km Well Diameter, D.5 m Well Roughness, ε.45 m 554