Russell James. D.S.I.R., Wairakei Private Bag, Taupo New Zealand. Flow, kg s RADIAL FLOW CALCULATIONS

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1 Proceedings Ninth orkshop Geothermal Reservoir Engineering Stanord University, Stanord, Caliornia, December 1983 SGP-TR-74 RADIAL FLO OF PRESSURED HOT ATER THROUGH NARRO CRACKS Russell James D.S.I.R., airakei Private Bag, Taupo New Zealand ABSTRACT Geothermal wells discharging hundreds o tonnes/hour o steam-water mixtures may be supplied at depth rom one very narrow crack o width 1 to 2 mm, or alternatively, rom some hundreds o hairline cracks. In the ormer case, turbulent low takes place out to tens o metres rom the well while the sum o rictional and kinetic pressure-drop indicates the lashing distance to be o the order o 1 cm rom the well wall or pressure-temperature equilibrium. However it is unlikely that equilibrium obtains because o the high water velocity (order o 1 m/s) near the well giving no time or bubble nucleation. Flashing and hence mineral deposition are thereore not at all likely in the crack but can occur within the well rom the crack horizon upwards. In the case o a multitude o ine cracks giving the same total low, streamline conditions prevail over the low path with the lash ront a metre or so rom the well, hence deposition is a possibility. INTRODUCTION The low o hot water to the eed level o geothermal wells may be considered as either through one crack or or the same low through a large number o much iner cracks. The latter condition is analogous to low through granulated beds but has the advantage o avoiding concepts o permeability expressed by units such as the darcy. The radial low o pressurised hot water has been studied, James, (1975a) using test results rom a well with a maximum discharge o 77 kg s-l (277 t h-l) with a crack width calculated at about 1.5 nun. Pressure ell rapidly within the crack when close to the well to below the value at which it would boil under stable equilibrium conditions. However, it was determined that the time interval or bubble nucleation was too short and hence that the luid remained as a single-phase until it burst into the well, at which point exposive generation o steam takes place ollowed by a continuous generation as the mixture ascends to the wellhead. This result is o some importance as i lashing commonly occurs within cracks, there would be a strong likelihood o deposition o minerals taking place which would be impossible - or extremely diicult - to remove, whereas deposition within the well. casing is a tractable condition even i not an attractive one. However, i the! low was distributed over a number o ine cracks, streamline (viscous) conditions would prevail over the low-path requiring a dierent calculation and in this case, the luid velocity would be severely restricted and the radius at which pressure alls to the boiling point would be increased. Both these eects lead to steam generation within the crack and hence a potential or mineral deposition would ncw exist. Because geothermal we1l.s commonly have sharply deined horizons o good inlow within the depth o interest with the remainder giving little, i any, eed over hundreds o metres o uncased hole, it appears that low rom a large number o minute cracks is rare compared with the case o one (or a ew) large cracks. It would be useul to study a well with a comparatively poor output (as a commercial proposition) and see which o these two conditions best apply. Fortunately such a well has been described recently by Menzies and others (1982) with a ew values o low and eed-horizon pressures reproduced below as Table 1. TABLE 1: Flow Characteristics o ell 43, Tongonan, Philippines. Flow, kg s RADIAL FLO CALCULATIONS 'b Feed horizon pressure, bar not available Taking the largest low with its ass ciated pressure rom the table; 28.8 kg s-' and 37.3 bar, we can calculate the crack width, t, rom the equation o James (ibid) making the assumption that turbulent low conditions operate rom the well outwards to a distance o -449-

2 at least a hundred well radii. This will be checked ater provisionally determining the crack width. The luid is considered as hot water using the published enthalpy o 127 Jg or this well giving an associated water temperature o C, speciic volume l and viscosity o.88 centipoise. Notation is given later. Using the metric orm, the pressure-drop along the radial low-path is: 1.85v,,.15 2 w V + t3 do.85 Po - Pb = [&I e may take it thereore that Equation (1) is applicable and hence we can estimate the eed horizon pressure Pb or various lows through the crack width above, as ollows: 12 - Pb = (22) (.6945) (1) For a well bore diameter o 22 mm and eed horizon pressure in the reservoir o 12 bar, we have: = [-:Tz]2i (.646) 2 2 ~ ' ~ ~ Hence Pb = 12 - [ (3) Solving,.15 (28.8) (.88) t 3 22ooee5 t = mm The changeover rom viscous to turbulent low is generally regarded as taking place at a Reynolds Number, Nr = 2, Perry (1963). where G t Nr = - and V Equation (3) is used to evaluate values o P or various values o w and these are plotte2 on Fig. 1 together with test results rom Table 1 and it is seen that good agreement is obtained. No sign o choking is indicated, as suggested by Menzies, et al. (ibid) and it appears that low progressively increases with lowering o eed horizon pressure. PRESSURE PROFILE TOARDS ELL To determine the luid pressure as it lows radially towards the well, we employ Equation (1) with d replaced by radius R in metres where d R = 2 (1) Flow is taken as w = 28.8 kg s-l and t =.646 nun with other actors as beore. Hence, 2 = t Pb = R'8 ] It 2 R (28.8) (.6945) 3.34 t3 (2 R)*85 ( '\ and For a low o 28.8 kg s and viscosity o.88 c'pise, the radius at the viscousturbulent interace is 26.4 mtres. Equation (2) is surprisingly independent o crack width and gives a radius to the viscous condition which is roughly equivalent numerically to the low-rate, and at a distance o 237 well bore radii. Equation (4) enables the pressure proile to be determined rom values o radius R and results are plotted on Fig. 2, where it is seen that pressure only starts to all signiicantly when within a radius o about 1 m rom the well centre-line. As the boiling pressure or water at 286.5OC is 7.14 bar, the associated radius is =.58 m. To evaluate the time taken in passing rom this radius to the well, we require the water velocity -45-

3 u = [&I I & TI [ v w v = - 2 s R t For w = 28.8, V = , t =.646 and assuming provisionally that there is no lashing (steam generation), 28.8 (1.3535) 9.61 u = = - m s 2 TI R.646 R, R =.1684 ' R = R - = U r =.11, r =.11 R. 6R 9.61 It is submitted that this is much too short a time to permit bubble nucleation, hence the water remains steam-ree beore it enters the well even though the pressure declines signiicantly below the boiling (saturated) value. HAT CRACK SIZE FOR COMPLETE VISCOUS FLO? From Equation (2) and taking R as equivalent to the well radius o.11 m we obtain;.11 = 4 TI (.88) hence w =.1216 kg s-l and or this low, viscous conditions apply over the whole lowpath. To determine the crack width, we employ the basic equation o James (1975b) in the metric orm and or Reynolds [e] Numbers less than 2. p w v In Po - Pb = 3 (5) t To obtain the identical low as beore, we require cracks each with.1216 the same pressure drop rom 12 to 37.3 bar. e take a value o the peripheral radius o 5 m approximating to hal the distance between wells = t = 5.88 (.1216) In mm t3 TO obtain the pressure proile with radial low towards tie well, we have: 12 - Pb = (.1216) In (.3) Pb = In 14 Equation (6) is used to plot the pressure proile or viscous low on Fig. 2 and can be compared with the case o turbulent low. Identical lows are assumed with one crack o width.64 mm passing a turbulent low o 28.8 kg s-' while or viscous low 237 cracks are required each o.3 mm width and passing.1216 kg s-l. For cracks narrower than.3 mn a larger number is required to sustain the same low, but the pressure proile remains the same. For all such viscous curves o Fig. 2, the pressure alls to the boiling value o 7.14 bar at a radius o 1.25 m rom the well centre-line and rom there to the well wall takes about.9 seconds or the.3 mm crack and much longer or narrow cracks. Hence lashing o steam is certain and the potential or mineral deposition exists, creating a 'skin' eect close to the well with increasing resistance and hence diminishing low with time. For viscous conditions and suicient cracks to give equivalent lows, a straight line relationship is obtained on Fig. 1 which can be compared with the curve derived or the same lows through one crack. Downhole measurements o lowing wells should permit dierentiation between these low types as suggested in James (1975a). CONCLUSIONS Probably the eed to geothermal wells is rom solo issures o a size greater than 1 mm or reasonable conmiercial discharges. For minute cracks o total. equivalent low, a very large number is required approaching thousands, and is analogous to low through granulated beds, which appears t:o be relatively rare, otherwise mineral scaling in the neighbourhood o wells, would be common in regions where the deep water has a high chemical content. Even wells which scale right down to the eed zone are rejuvenated ater reaming, indicating that solids are not deposited within the rock ractures. I the opposite were true, geothermal science would be aced with potentially serious problem o descaling the matrix

4 REFERENCES NOTATION Menzies, A.J., Gudmundsson, J.S. and Horne, R.N. 1982: Flashing Flow in Fractured Geothermal Reservoirs. Proc. 8th orkshop, Geothermal Reservoir Engineering, Stanord University, Caliornia, U.S.A. James, R. 1975a: Drawdown Test Results Dierentiate between Crack Flow and Porous Bed Permeability. 2nd U.N. Symp. Development and Use o Geothermal Resources. San Francisco Caliornia, U.S.A. James, R. 1975b: Optimum ell Spacing or Geothermal Power. 2nd U.N. Symp. Development and Use o Geothermal Resources. San Francisco, Caliornia, U.S.A. Perry, J.H. 1963: Chemical Engineers' Handbook. McGraw-Hill, U.B.A. d G Nr PO % r R t u Diameter, mm. -2 Mass-velocity, kg m s Reynold's Number (non-dimensional). Reservoir pressure (no-low) at eed horizon, bar. Pressure at eed horizon, bar. Radius o well, mm. Radius, m. Crack width, mm. Velocity o hot water, m s. -3 Speciic volume-p hot water, kg m. Flow-rate, kg s. Viscosity o hot water, centipoise. 3 test results rom Table 1 I turbulent low 2 w kg m-' 1 Figure 1. Flow-rate versus pressure at eed horizon. streamline (viscous) IO 12 Pb bar -452-

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