Proceedings GeothermalWorkshop 2001

NUMERICAL MODELLING OF GEOTHERMAL RESERVOIRS CONTAINING FLUIDS NEAR THEIR THERMODYNAMIC CRITICAL POINT

S. J. & A. WATSON2 'Department of EngineeringScience,University of Auckland 'Geothermal Institute,University of Auckland

SUMMARY- This work investigates natural convection in geothermal reservoirs containing water near the thermodynamic critical point. The peculiarity in pure water properties near the critical point and the extent of the critical region for supercritical pressures are shown. PHOENICS commercial computational fluid dynamics (CFD)was used to simulate natural convection porous media with near critical properties, which are highly temperature dependent. These near-critical properties were modeled by simple algebraic representations. Speculations on the implications of the conditions in real geophysical systems are then discussed in view of the results obtained.

1. INTRODUCTION the critical pressure of pure water. The pure water critical conditions are seldom met, and the The point where the liquid saturation line and temperatures reached are much less except in a vapour saturation line for a pure substance meet few cases. However, reservoir conditions of 380 on a P-v diagram is called the thermodynamic "C and 220 bar abs. have been reported at critical point. At pressures above this point there Nesjavellir field in Iceland by Steingrimsson et is no clear division between the superheated al. (1990). At the Kakkonda field in Japan a vapour region and the compressed liquid region. temperature higher than 449 was reported by Phase changes from liquid to vapour or vice versa Yano and Ishido (1990). The existence of near- occur at pressures below this critical point. For critical conditions in geothermal reservoirs has pure water the critical pressure and temperature also been inferred from the study of fluid are 221.2 bar abs. and 374.15 "Crespectively. inclusion samples (Reyes, 1993).

The critical conditions have attracted the attention The presence of dissolved minerals and gases in of researchers from different disciplines over the geothermal water has the effect of shifting the many years. Van der Waals proposed an equation critical temperature and pressure to higher values of state for the critical region in 1873. More and reducing the size of property variation peaks recently interest in the near-critical region came (White et al. 1971) and (Bodnar and Costain, from several trends in engineering. Some of the 1991). NaCl is the main dissolved mineral in early applications investigated were the geothermal fluids, while CO, is the main gas. NaCl development of fossil fuel power plants for near- increases the temperature and pressure of the critical conditions and also the use of supercritical critical point compared to pure water (Straus and pressure water as a coolant in nuclear reactors. Schubert, 1977). The presence of NaCl also Helium at near-critical conditions is used as a increases electrolyte concentration, increasing the coolant for superconductors. In relation to deep solubilityof silica and shifting the quartz solubility geothermal reservoirs, areas of interest are the maximum to higher temperature and pressure, cooling of magma bodies, formation of seismic creating an sub-system that further boundaries, and ore deposition. Norton and influences the critical point (Johnson and Norton, Knight (1977) and Dunn and Hardee (1981) have 1991 and Bodnar and Costain, 1991). CO, on the drawn attention to the possible significance of the other hand increases the pressure but reduces the near-critical region in geophysical fluid flow and temperature at which the critical point occurs heat transfer. (Bodnar and Costain, 1991). These shifts increase the likelihood of zones of near-critical fluid having The occurrence of geothermal prospects that can significant size as the concentration of solution be developed for major energy extraction is varies with depth. restricted to locations with plate tectonic activity, where the hot rocks and fluid are at accessible Whilst drilling into a near-critical reservoir would depth. At present, wells drilled for geothermal produce interesting results, the focus of this paper power development reach maximum depths of the is on natural convection at greater depths in the order of 3500 m. Assuming hydrostatic pressure crust. The physical processes by which localized and a water level at the ground surface, the heat flows reach the surface are not yet fully maximum pressure that could be reached in such understood. High temperature rocks are a well is approximately 350 bars, which is above anticipated to be plastic and impermeable, and

237 vertical heat fluxes therefore generally low. It is important to note that the peaks in properties critical conditions of the type examined in this also occur above the critical pressure, but to a paper may allow heat to flow laterally to locations lesser extent, and the pressure and temperature of from which it can escape vertically. these is called the pseudo-critical point. Significant effects extend to about 1.2 times the critical 2. OBJECTIVES pressure (Hall, 1971).

The aim of this work is to investigate the form 10000 2600 and behaviour of geophysical flows of near- 2500 critical fluid using computer modeling. More 2400 specifically, this study concentrates on natural convection in porous medium of fluids close to 1000 2300 but just above the critical pressure, where the 2200 property variations with temperature are 2100 extremely large. The fluid is contained in a 2000 rectangular 2-D cell with isothermal bottom and top, the bottom being hotter, and adiabatic 1900 vertical walls 1800 10 1700 3. CRITICAL PROPERTIESOF WATER

At the critical point, pure water exhibits extreme variations of properties with temperature and pressure, but more strongly with temperature. Figure 2. Specific heat capacity and Specific These property variations have the potential to enthalpy of pure water at the critical pressure modify various common modes of heat transfer (221.2 bar), based on IAPWS-IF97. and create unusual flows (Hall 1971). Numerous equations of state for different fluids have been 60 proposed since Van der Waals, but they all fail to 800 represent the critical region as a continuous 55 function of pressure and temperature. 50 The International Association for the Properties of 600 Water and Steam Industrial Formulation 45 E 500 previously known as 1967 (International 40 has produced accurate properties. 30 200

100 25

1.00

Figure 3. Thermal Conductivity and Dynamic viscosity of pure water at the critical pressure (221.2 bar), based on 200

4. PREVIOUS STUDIES ON POROUS MEDIA

The only laboratory experiment to examine natural convection of near-critical pressure fluid Figure 1.Coefficientof thermal expansion and in porous media is that of Dunn and Hardee density of water at the critical pressure (221.2 (1981). The experimental apparatus consisted of a bar), based on 1litre cylinder with a length to diameter ratio of 7 packed with silica sand and saturated with pure Using these with constant pressure of 221.2 bar water. The cylinder was fully insulated and the abs. (the critical pressure) localised peaks in the outer sidewall was kept at constant temperature coefficient of thermal expansion (Figure while a platinum wire heated the center line. A specific heat capacity (Figure 2) and thermal hand pump maintained the pressure between conductivity (Figure 3) are revealed with change in 245 bar and temperature measurements were temperature. A significant drop in density (Figure taken by thermocouples embedded in the sand 1) and dynamic viscosity (Figure 3) and an radially around the platinum wire. increase in specific enthalpy (Figure 2) also occur.

238 The measured heat transfer rate near the critical near-critical region described in the point (370-380 showed an increaseof about 70 and the 1967 IFC version give good smooth times the rate at room temperature. property variations in the near-critical region with full representation of the peaks, but the equations Cox and Pruess (1990) numerically simulated this are very complex. Furthermore there is the experiment using the MULKOM geothermal question of dealing with small variations about reservoir simulator and calculated an enhancement high standing values, which was discussed by Cox in heat transfer rate of only 5 times that of pure and Pruess (1990) and is a problem for numerical conduction, even though the permeability used was simulation. To avoid these problems we have 20 times greater than that reported in the "invented" a fluid with property variations with laboratory experiment. This contradiction between temperature to suit our investigation. The invented experiment and numerical simulation has not been fluid has the (constant) properties of atmospheric explained. There is no doubt that the experimental pressure water with the exception of and Cp arrangement may not have allowed clear which are anticipated to dominate convection in conclusions, and experiments with near-critical the near-critical region. The real variation of fluid are difficult because of property variations and with temperatureis much greater than with with pressure and temperature. On the other hand pressure and these parameters depend only on the authors' preliminary experience with temperature in the invented fluid. MULKOM led them to the belief that it was not ideal for single-phase natural convection studies. The assumed is shown in Figure 4; when solving with a fine the discontinuity in AND HEAT TRANSFER IN POROUS is insignificant. MEDIA 0.0020 I I 1000 The present work was carried out for 0.0018 998 0.0016 dimensional rectangular geometry, for which the 996 0.0014 governing equations are conservation of mass, E 994 momentum and energy:- E 0.0012 0.0010 992

988

0.0002 986 0.0000 984 20 25 30 35 40 45 Temperature C)

Figure 4. Typical assumed variation of expansion coefficient with temperature and the resulting variation of density with temperature where: The form assumed allowed examination of effects of different peak properties and amplitudes. The equations are:

= =Constant for c T and T (5)

These equations were solved using the = a + for T c T, and T, T (6) PHOENICS simulator. Experimental measurements in cellular convection of ordinary where a and b are specific to the temperature fluids were replicated using this program for range. verification Zarrouk Neglecting variations with pressure, the definition 6. REPRESENTATION OF of thermal expansion: CRITICAL PROPERTIES FOR THIS STUDY

The central problem in understandingnear-critical crustal flows is to learn what happens to a fluid that rises into a region of near-critical pressure and allows integration to give a density variation of temperature. The properties of pure water in the general form:

239 T )+a(T ) where a, b, and can be expressed in terms

Solutions were carried out with different where: is the residual of variable amplitudes of the peak, characterisedas . of solution and is the relaxation parameter. By careful numerical investigation on the solution A similar form has been assumed for Cp, and the controls (linear relaxation, and number of enthalpy variation with temperature that results iterations) and from experience in watching the can similarly be at by integration (Figure solution develop and how the residuals decrease and also from mesh refinement study, we believe that the results obtained so far are representative (9) of the near-critical region, for the given assumptions. for If we consider - -), 2 where is the stream function, and where e and fare again specific to the temperature represent the amplitude of the peak in those range. Figure 5 shows Cp and the corresponding properties, then the value of Ra based on plate enthalpy for = temperature difference cannot be used alone to distinguish flow regimes. So the onset of unsteady 45.0 convection probably depends on the amplitude as 40.0 well as the width of 'the property peaks.

35.0 300.0 8. DISCUSSION OF RESULTS 30.0 25.0 The implications of natural convection in porous 20.0 media with near-critical fluid for geophysical systems have led to several speculations. Straus 15.0 E (1977) E 100.0 and Schubert noted that these conditions 10.0 might beneath geothermal fields, most likely 5.0 above shallow magma bodies in the crust. Tc T2 Nevertheless as the magma continues to rise 0.0 0.0 towards the surface, maintaining a critical 20 50 25 35 40 45 pressure will become unlikely especially in an Temperature(deg C) unconfined geothermal system under hydrostatic pressure, and as two-phase conditions form (White Figure 5. Typical assumed variation of specific 1971 1981). heat capacity with temperature, for and Dunn and Hardee Dunn and Hardee (1981) suggested that the intense convection could significantlyweaken a porous or a fracture-controlled permeable medium, due to 7. enhanced rock dissolution. The solubility of some 5 species becomes very high at near-critical For the property variations of Figures 4 and with conditions; this is the basis of supercritical and the results show separation in process engineering. The increase in an unsteady fluid-motion taking place (Figure 6), permeability due to rock dissolution would also resulting in an unsteady temperature distribution. mean an increase in the ability of the geothermal Near-critical temperature fluid occupies most of system to transfer energy and Costain the space. Though the solution is numerically 1991). stable, it is found that no steady state is reached and the patterns continue to change. Figure 7 A question that arose at the outset of this work shows the results after seconds where the was whether an isolated layer of convection cells critical specific heat capacity and density is could occur due to high and following the temperature distribution. Increasing ratios, between plates that were spaced the ratios of both and caused far apart and had a small temperature difference, numerical instability initially. The decision of so that the Ra value based on the plate temperature whether the unsteadiness is numerical instability was less than This question has not yet been or a real fluid flow and heat transfer effect was This crucial. PHOENICS uses the whole field residuals answered. might be an important issue in method for convergence. geophysical terms. In the Taupo Volcanic Zone of New Zealand, there are many geothermalresources lying close together but with thermodynamic, electrostatic, and transport separate resistivity boundaries. There has been properties of in the critical region. American speculation that these are the result of cellular of Science, Vol. 291, pp. 541-648. convection above a deep “hot plate” 1993). The mechanism for lateral heat flow to McKibbin R. and A. (1993) Modelling create such a plate has not been satisfactorily the phase boundaries and fluid properties of the explained. An isolated layer of counter-rotating system at high temperature and convection cells provide a mechanism for pressure. Proceedings the Geothermal lateral heat flow. It is possible that the magma Workshop. intrusions are localised effects and the near- critical regions provide the means of expanding Norton D. and Knight J. (1977). Transport the effects of magma intrusionslaterally. phenomena in hydrothermal systems: Cooling plutons. American Jnl of Science, Vol. 277, pp. The heat transfer coefficient in the near-critical 937-981. zone (the effective conductivity) be very high and with the high specific heat would keep Reyes A. G.(1993). Petrology and geochemistry the zone at almost uniform temperature, since the of Alto Peak, a vapour-cored hydrothermal vertical heat transfer rate is restricted. This has system, Leyte province, Philippines. Geothennics been demonstratedin our work, Figure 6. 22, 479-519.

From Figure 6, it is clear that the critical zone is Steingrimsson B., Gudmundsson A., Franzson H. confined within the upper and lower temperature and Gunnlaugsson E. (1990) Evidence of a boundaries allowing lateral expansion only. supercritical fluid at depth in the Nesjavellir field. Whether it is possible for cellular convection to Proc. Stanford Workshop. take place below and above the critical zone is still used open to some question, since the geometry is Straus J. M. and Schubert G. (1977). Thermal not large enough to show such effectclearly. convection of water in a porous medium: Effects of temperature and pressure-dependent 9. CONCLUSIONS thermodynamic and transport properties. Jnl GeophysicalRes.,Vol. 82, pp. 325-333. It appears that the cellular convection does occur in fluids with near-critical properties, Watson A. (1996). Some possible characteristics but the convection is always unsteady, of geothermal reservoirs containing near-critical probably due to the very rapid change of pressure fluid. Proc. PNOC Conference, Manila, property with temperature. 1996.

It has not been fully proven by this study, but White D. E., Muffler L. J. P. and Truesdell A.H. there are indications that the critical property (1971). Vapour-dominated hydrothermal systems region exists as a thin layer with the ability to compared with hot-water systems. Economic transfer heat laterally. Geology,Vol. 66,pp. 75-97. REFERENCES 10. Yano Y. and Ishido T. (1990) Numerical investigation of pressure transient responses of a Bodnar R. J. and Costain J. (1991). Effect of mass well penetrating a deep geothermal reservoir at varying fluid composition on and energy super-critical conditions. Proc. Stanford transport in the earth crust Geoph. Res let, Vol. Workshop. 18, 983-986. Zarrouk S. J., Watson A. and Richards P. J. Cox B. L. and Pruess (1990). Numerical experimentation on convection heat transfer in The use of PHOENICS computational fluid dynamics code for geothermal water-saturated porous media at near-critical reservoirs containing fluids near their conditions. Transport in Porous Media., Vol 5, thermodynamic critical point. Proc. 24” Stanford 299-323. Workshop. Dunn J. C. and Hardee H. C. (1981). J. of geothermal zones. Jnl and Zarrouk S. Numerical solution near-critical natural convection in porous media Geoth. Res., Vol. 11, pp.189-201. with reference to geothermal reservoirs. ME Thesis (Mechanical Engineering), University of Hall W.B. (1971). Advances in heat transfer. Auckland. Irvine, Jr. T. F. and P. J. Academic Press, New York, Vol. 7, pp. 1-83. Nomenclature Johnson J. W. and Norton D. (1991) Critical a,b coefficientsof variation phenomena in hydrothermal systems: state, Cp specific heat at constantpressure

241 coefficients of h variation Isothems acceleration due to gravity h specific enthalpy K permeability k thermal conductivity P pressure R residuals Ra y-Rayleigh number T temperature Streamlines t time V velocity in y direction (horizontal) W velocity in zdirection (vertical) spacial coordinates

themal expansion coefficient porosity dynamic viscosity Specificheat capacity density general field variable

critical - f fluid m mixture n time step Density r rock initial state

Isotherms Streamlines

Afer Figure 7. Detailedresults for after

Afer

Figure 6.Transient results for =10

242