Liquid Holdup in Geothermal Wells

Transcription

1 Proceedings World Geotheral Congress 005 Antalya, Turkey, 4-9 April 005 Liquid Holdup in Geotheral Wells Sabodh K. Garg, John W. Pritchett, and Jaes H. Alexander Science Applications International Corporation (SAIC), 1060 Capus Point Drive, San Diego, CA 911, USA Keywords: Holdup correlation; pressure and teperature profiles; well siulators; siulation; two-phase flow; spinner ABSTRACT Siulation of two-phase flow in geotheral wellbores requires use of epirical correlations for liquid holdup and for friction factor. Use of currently available correlations often yields widely differing results for geotheral wells. A new liquid holdup correlation is devised for cased wellbores using high-quality discharge and downhole pressure and teperature data fro flowing geotheral wells. The latter dataset encopasses a wide range of wellbore diaeters, discharge rates and flowing enthalpies. The easured wellhead pressures for wells in the dataset display excellent agreeent with the pressures coputed by using the new holdup correlation. 1. INTRODUCTION An ability to predict both the quantity of fluid that can be produced and its therodynaic state (pressure, teperature, enthalpy, gas content, salinity, etc.) is essential for estiating the total usable energy of a geotheral resource. Nuerical reservoir siulators can be utilized to calculate the therodynaic state of the fluid at the underground feed-zone(s) at which the fluid enters the wellbore. The coputation of the well-head fluid properties fro a given underground state (or vice-versa) requires the use of a wellbore siulator. Treatent of two-phase flow in a wellbore requires use of epirical correlations for liquid hold-up and friction factor. Because of slip between the gas and liquid phases, the flowing gas quality Q f is generally different fro the in situ gas quality Q s. The liquid hold-up correlation provides a relationship between Q f and Q s. Alost all of the existing holdup correlations (see e.g. Ansari et al., 1994; Aziz et al., 197; Beggs and Brill, 1973; Duns and Ros, 1963; Hadgu, 1989; Hagedorn and Brown, 1965; Hughark, 196; Hughark and Pressburg, 1961; Orkiszewski, 1967) are based on flow in two-phase petroleu (oil and gas) systes. At present, there does not exist a sufficient basis for selecting one or another of these correlations to siulate two-phase flow in geotheral wellbores. Utilization of different correlations very often yields widely differing results (see e.g. Finger, et al., 1999). In this paper, we describe the developent of new hold-up correlations for geotheral applications using data fro flowing geotheral wells. To support this research work, proprietary downhole logs and other required data were supplied by Unocal, Caithness, and various Japanese developers. As a result of a detailed exaination of these well data, we identified over 30 wells with high quality discharge (ass discharge rate and enthalpy) and downhole pressure and teperature data. The data set encopasses a wide range of discharge enthalpies (i.e. oderate enthalpy 1 wells producing fro liquid feedzones, and wells with enthalpies approaching the enthalpy of saturated stea), and casing diaeters (ID s ranging fro 100 to 384 ). As far as fluid coposition is concerned, the data set leaves soething to be desired. The salinity and noncondensable gas content of ost of the wells in the data set are less than 1.5% and 1% (ass fraction of the produced fluid), respectively. In any event, this data set is einently suitable for developing a new epirical liquid hold-up correlation for geotheral wells. Our ethodology for developing a hold-up correlation is outlined in Section. The downhole pressure/teperature profiles were siulated using an existing wellbore code with an adjustable hold-up correlation. The results of these siulations were eployed to generate a ulti-paraeter data set. The latter data are used in Sections 3 and 4 to relate flowing gas quality to other paraeters. Spinner data are eployed in Section 5 to provide an independent check on the holdup correlations developed in Sections 3 and 4.. FLUID FLOW IN GEOTHERMAL WELLS The pressure drop associated with two-phase fluid transport in a geotheral well represents the cobined effects of friction, acceleration, and the loss of elevation. While the pressure drop due to acceleration is usually sall in singlephase liquid flow in a pipe, it is very often the ost iportant coponent in two-phase flow. The fluid flow in a geotheral well is not aenable to strict analytical treatent. For two-phase flow, it is necessary to suppleent ass, oentu and energy balance relations by epirical correlations for (1) friction factor, and () liquid hold-up. In the following, a wellbore siulator, incorporating an existing friction factor correlation and an adjustable liquid hold-up correlation, is used to atch the downhole pressure profiles in flowing wells. The siulation results are eployed to generate a ulti-paraeter (flowing quality, static quality, in situ liquid and gas volue fractions, gas and liquid viscosities, etc.) data set. The latter data sets fors the basis for the developent of a new liquid hold-up correlation. The downhole pressure (and teperature) profiles in the cased portion of flowing wells have been siulated using a specially odified version (see below) of the wellbore coputer siulation progra WELBOR (Pritchett, 1985). The WELBOR code treats the steady flow of liquid water and stea up a borehole. The user provides paraeters describing the well geoetry (inside diaeter and angle of deviation with respect to the vertical along the hole length), a stable foration teperature distribution with depth, and an effective theral conductivity as a function of depth representing the effects of conductive heat transfer between the fluid in the wellbore and the surrounding foration. For boreholes with two-phase flow at the botto of the cased portion, the fluid state is prescribed by specifying flowing pressure, flowing enthalpy, salinity, and gas content.

2 In two-phase water/stea flow, pressure and teperature are not independent of each other. For any reasonable value of effective theral conductivity κ, the downhole flowing enthalpy ay be adjusted to yield the appropriate pressure, and hence teperature, distribution in the wellbore, and flowing wellhead enthalpy. Matching the pressure/ teperature distribution in the wellbore and flowing wellhead enthalpy does not constrain the heat loss and downhole enthalpy. Since the flowing downhole enthalpy is not a easured quantity, it is not possible to deterine a unique value for heat loss in the presence of two-phase flow. It is, therefore, appropriate to eliinate effective theral conductivity as a free paraeter, and use a constant value for κ (= 4 W/- C). For the present study, the frictional pressure gradient is coputed using the Dukler I correlation (Dukler et al., 1964), and a user prescribed roughness factor, ε. The roughness factor, ε, ay vary with depth. For ost of the pressure profiles considered herein, the roughness factor was assued to be zero. In a few cases, it was found necessary to use a non-zero value for ε. The relative slip between the liquid and gas phases is treated in WELBOR using a odified version of the Hughark liquid holdup correlation (Hughark, 196). The slippage rate ay vary between the value given by the Hughark correlation and no slip at all, according to the value of a user specified paraeter, η, which varies between zero (no slip) and unity (Hughark). For the present application, the WELBOR code was odified so as to allow η to vary as a function of depth. For all of the pressure profiles considered herein, it was found that at ost two values of η (and a sall transition zone in between) were required to produce a satisfactory atch between the easured and coputed pressures. Given the fluid state at the botto of the cased interval and the ass discharge rate, a wellbore siulator such as WELBOR can be used to copute the fluid state along the wellbore and at the wellhead. The principal paraeters that ay be varied to atch the easured conditions along the wellbore (pressure and teperature) and at the wellhead (pressure, teperature, stea and liquid flow rates, liquid salinity, gas content of stea) are (1) flowing enthalpy, salinity and gas content at the botto of the cased interval, () holdup paraeter, η, and (3) interior roughness factor, ε. To illustrate the coputational procedure, it is useful to consider Unocal well A-4 (Garg and Pritchett, 001). Well A-4 is cased and ceented to a depth of TVD (901.6 MD). The following well geoetry is assued for the cased section of well A-4: Measured Depth (eters) Vertical Depth (eters) Angle with Vertical (Degrees) Internal Diaeter () The easured pressures in the flowing well are in good agreeent with the saturation pressure for pure water corresponding to the easured teperatures (Garg and Pritchett, 001). A pressure of ~3.49 bars (taken as the average of easured and saturation pressures) was recorded in the flowing well at TVD. The reported discharge rate and wellhead enthalpy were 135 kg/s and 1089 kj/kg, respectively. Total dissolved solids content of the separated liquid was pp; the non-condensable gas content of the stea was 0.68%. The stable foration teperature (Garg and Pritchett, 001) was approxiated by the following teperature distribution using linear interpolations between tabulated data. Vertical Depth (eters) Teperature (Degrees Celsius) The best atch to the downhole pressure profile and wellhead fluid state (pressure, enthalpy, salinity, gas content) was obtained using the following values for unknown odel paraeters: Flowing enthalpy = 110 kj/kg at TVD Fluid (liquid + stea) salinity = 0.01 kg/kg at TVD Fluid (liquid + stea) gas = kg/kg content at TVD Hughark paraeter, η = 0.09 for depths < 350 = (depth 350) for 350 < depth < 400 = 0.40 for depths > 400 Roughness factor, ε = 0.00 for all depths The coputed pressure profile is copared with the easureents in Figure 1; the agreeent is excellent. The coputed fluid state at the wellhead (fluid enthalpy: 1089 kj/kg, liquid phase salinity: 14,300 pp; stea phase gas content: 0.67%) is very close to the easureents. Vertical Depth, eters Discharge Profile A-4 Saturation pressure Coputed (best atch) Figure 1: Pressure profile (triangle) recorded in discharging well A-4. The squares indicate saturation pressure corresponding to local easured teperature. The solid line is the coputed pressure profile. An essentially identical procedure was used to fit the downhole pressure and wellhead fluid state easureents for all of the wells in the dataset (see Garg et al., 004 for details). The data for the Unocal wells (and presuably the

3 single Caithness well) were obtained while these wells were discharging in a ore or less stable anner. Available data for these wells include the NCG and salt content of the discharge strea. By way of contrast, data for the Japanese wells were obtained during short ter discharge tests. No NCG and salinity easureents are available for the production fluid fro these short ter tests. Also, discharge rate and wellhead enthalpy easureents fro the short ter tests ay not be very accurate; these errors are likely to be ost probleatical for wells with very low discharge rates. Although flow data for the Japanese wells in the dataset are liable to be less accurate than for the Unocal wells, it was decided against discarding these data in that the Japanese well test data are typical of relatively high quality easureents taken during the exploration phase. The results of these downhole pressure/teperature siulations (Garg et al., 004) were used to define the fluid state and associated quantities (e.g. liquid and gas velocities) in the cased section of all the wells in the dataset. Soewhat arbitrarily, it was decided to use 1 equally spaced points along each downhole profile to create a dataset (see Table.1 in Garg et al., 004 for an exaple) for forulating a new holdup correlation. 3. A NEW HOLDUP CORRELATION Duns and Ros (1963) suggest that the various flow regies that accopany two-phase flow in wells can be divided into three ain regions depending on the gas throughput (Figure ). The axes in Figure denote the non-diensional liquid and gas velocity nubers: Liquid velocity nuber, Gas velocity nuber, N = S v ( ρ / gσ) l l l l N = S v ( ρ / gσ) g g g g Here vl ( v g ) is the liquid (gas) velocity, Sl ( S g ) is the in situ liquid (gas) volue fraction, ρl ( ρ g ) is the liquid (gas) density, g is the acceleration due to gravity, and σ is the surface tension. Region I has a continuous liquid phase, and contains bubble flow, plug flow and part of froth-flow regies. Liquid and gas phases alternate in region II covering the slug flow and reainder of the froth-flow regies. Region III is characterized by a continuous gas phase, and contains the ist-flow regie. Data fro two-phase geotheral wells are shown as diaonds in Figure. Although the geotheral data lie in all the three regions, the bulk of these data are contained in Region II. It appears fro Figure that relatively high liquid velocities characterize geotheral wells such that only froth-flow (regions I and II) and ist-flow (region III) are encountered in geotheral wells. Once the geotheral fluid starts flashing in the wellbore, the gas (and hence liquid) velocity increases rapidly. Thus, bubble flow, plug flow and slug flow regies, if present at all, are likely to be confined to a narrow depth range and difficult to observe. Because of the liited range of flow regies, it should be possible to describe two-phase flow in geotheral wells by a single (or at ost a two-part) holdup correlation The flowing quality Q f (=gas flow rate/total flow rate) is defined as follows: Q f ASgρgvg = (1) M N l BUBBLE FLOW PLUG FLOW FROTH FLOW REGION I HEADING SLUG FLOW REGION II REGION III MIST FLOW N g TRANSITION Figure : Two-phase fluid flow regies according to Duns and Ros (1963). Also shown (as diaonds) are the data fro geotheral boreholes (see Section above). where A is the internal cross-sectional area of the pipe, and M is the total ass flow rate. Bankoff (1960) derived a relation for flowing quality Q f that is equivalent to: Q f Qs = [1 Q ] K + Q [1 ρ (1 K) / ρ ] s s l g where Q s is the in situ (or static) quality, ρ is the ixture (gas plus liquid) density, and K is a flow paraeter (see below). () Sgρg Qs = (3) ρ ρ = S ρ + S ρ (4) l l g g For the case of hoogeneous (i.e. no slip) flow, the in situ quality Q s is equal to the flowing quality Q f ; furtherore, flow paraeter K is identically equal to unity. In general, one would expect the gas phase to rise ore rapidly in the well than the liquid phase due to buoyancy; this iplies that Q Sg f Q (5a) s K 1 (5b) The two-phase flow in a well is influenced by buoyancy, inertial, viscous and surface tension forces (Bankoff, 1960). Based on diensional arguents, Hughark (196) concluded that the flow paraeter K ight be expected to depend on the flowing liquid volue fraction Y l, and Reynolds (Rn), Froude (Fr), and Weber (We) nubers. Sv l l Yl = ( Slvl + Sgvg) dwm Rn = Aµ M Fr = dwa gρ dwgρgfr We = σ (6a-d) 3

4 Here d w is the inside well diaeter, σ is the liquid surface tension, and µ is the ixture viscosity. µ = S µ + S µ (7) l l g g In Equation (7), µ ( µ ) denotes the liquid (gas) viscosity. l g Mixture (liquid plus gas) density ρ and ixture viscosity µ can be defined in a nuber of ways (see e.g. Hughark, 196; Dukler, et al., 1964); in the following, the expressions given by Hughark, Equations (4) and (7), will be used. The latter definitions for ρ and µ are different than those used by Dukler, et al. (1964). To find a correlation for K, it is assued that K can be expressed as a function of a single variable Z: K = K( Z( Rn, Fr, Y, We, S )) (8) Hughark (196) investigated the dependence of Z on Rn, FrY, l, and We, and found that Z (and hence K) did not appreciably depend on the Weber nuber (We ). The authors have independently confired the latter result using data for geotheral wells (Section ). Consequently, Z is assued to depend on only four variable, i.e., Rn, FrY, l and S l ; note that Hughark (196) did not account for the dependence of Z on S l. Following Hughark (196), we introduce a particularly siple relationship for Z: α β γ ω l l l Z = Rn Fr Y S (9) where α, βγ,, and ω are as yet undeterined constants. Deterination of the exponents in Equation (9) is straightforward when the functional relationship between K and Z is known; in this case, exponents can be estiated by iniizing the variance between the calculated (i.e. fro the functional relationship K-Z) and easured (i.e. those derived fro fitting the flow data in Section ) values for K. Unfortunately, the functional relationship K(Z) is unknown, and ust be deterined fro the dataset. To deterine the exponents in Equation (9), we introduce a nonparaetric easure of variation. Given a candidate set of exponents, we calculate Z for each point in the dataset for geotheral wells (Section, and Figure ). Next, we sort the dataset in order of increasing Z, and calculate a pseudo-variance S.V. as follows: l n 1 i+ 1 i (10) i= 1 SV.. = ( K K ) Here K i denotes the value of K corresponding to Z i, n is the nuber of points in the dataset, and Z n is the largest value of Z. The exponents in Equation (9) are obtained by iniizing this pseudo-variance. The function K(Z) looks like a hyperbola with K = 1 as an asyptote (see e.g. Figure 3). Although a hyperbola for K(Z) can be ade to yield a pseudo-variance that is close to the iniu, a ore general functional for for K(Z) is needed in order to iprove the fit in regions of Z that ake little contribution to the pseudo-variance. To verify that the holdup correlation (i.e. K(Z) relationship) developed herein can be used to siulate two-phase flow in geotheral wells, a special version of WELBOR code was created; this version was configured to use the K(Z) relation K Z Figure 3: A plot of K versus Z. Data points (+), i.e. K versus Z values, in the figure were obtained by iniizing the pseudo-variance. Z is defined by Equation (9) with α = , β = , γ = , and ω = 0.1. The solid line denotes an analytical fit to the K-Z data. (solid line) shown in Figure 3. The latter version of WELBOR was eployed to siulate flow data for all the wells in the dataset. Except for the holdup correlation, these siulations utilized the paraeters used in the calculations described in Section. The coputed results for wellhead pressure for all the high ass velocity (ass velocity = total discharge rate / pipe cross-sectional area > 687 kg/s-. The low ass velocity profiles are considered separately in Section 4.) profiles are copared with data (i.e. wellhead pressures coputed by atching downhole pressure profiles in Section ) in Table 1. The K(Z) fit yields satisfactory agreeent between data and coputed wellhead pressures (Table 1); the root-eansquare error is only 0.67 bars. It is apparent fro Table 1 that the above-derived correlation for K(Z) leads to choking for three pressure profiles (KE1-4a, KE1-4b, KE1-19Sa). Well KE1-4 is a sli hole (internal diaeter = 10. c), and the downhole profiles were recorded during a short-ter discharge test. The reported discharge rates for KE1-4 ay be in substantial error. The coputed axiu discharge rate for KE1-4 is around 6.9 kg/s, which is within the likely error band for the reported discharge rate for KE1-4. For KE1-19Sa, the coputed value for the axiu discharge rate (3.0 kg/s) is within 4% of the noinal discharge rate (3.9 kg/s). Thus, it can be concluded that the K(Z) fit gives acceptable results for these three profiles as well. As entioned above, downhole pressure profiles for all the high ass velocity cases in the dataset (Table 1) were siulated using the correlation for K(Z). In ost cases, the latter coputed profiles are in good agreeent with (1) easureents, and () calculated profile (best-atch) using an adjustable holdup correlation (Section ). A typical exaple (well A-4) is shown in Figure 4; the coputed pressure profile for well A-4, shown as a dashed line in Figure 4, is in excellent agreeent with the easureents. 4. HOLDUP CORRELATION FOR LOW MASS VELOCITY The correlation for K(Z) developed in the preceding section was found to yield satisfactory results for all the high ass velocity (ass velocity > 687 kg/s- ) profiles in the 4

5 dataset. The coputed pressures for low ass velocity 0 Discharge Profile A-4 Table 1: Coparison of wellhead pressures for high ass velocity profiles (ass velocity > 687 kg/s- ) coputed using the correlation for K(Z) with data. Well P-data P-cop δp *** A A A A A A A A A A A A A A A A B C KE1-4a * NA KE1-4b NA KE1-17a KE1-19Sa 3.53 ** NA KE1-19Sb KE1-b GH-11a GH-11b GH-0a GH-0b GH-0c Total (6 profiles) 1.11 * At 6.9 kg/s, p =.31 bars; 6.8 kg/s, p =.66 bars; 6.5 kg/s, p = 3.8 bars ** At 3.0 kg/s, p = 3.66 bars. *** δp = P-data P-cop (ass velocity < 650 kg/s- ) profiles are, however, too high, and iply that the correlation for K(Z) would need to be odified for low ass velocity. Attepts to develop a single correlation for both the high and low ass velocity cases by including an additional variable (i.e. ass velocity) in the relation for Z (Equation 9) were unsuccessful. Accordingly, it was decided to consider the low ass velocity profiles separately. The K(Z) correlation for the latter case is shown in Figure 5. The coputed results for wellhead pressure for all the low ass velocity (ass velocity = total discharge rate / pipe cross-sectional area < 650 kg/s-.) profiles are copared with data (i.e. wellhead pressures coputed by atching downhole pressure profiles in Section ) in Table. The K(Z) fit shown in Figure 5 yields satisfactory agreeent between data and coputed wellhead pressures (Table ); the root-ean-square error is 0.73 bars. Vertical Depth, eters Saturation pressure Coputed (best atch) Coputed (holdup correlation) Figure 4: Pressure profile (triangles) recorded in discharging well A-4. The squares indicate saturation pressure corresponding to the local easured teperature. The solid line is the coputed pressure profile using an adjustable holdup correlation (see Section for details). The coputed pressure profile using the correlation for K(Z) is shown as a dashed line. Figure 5: A plot of K versus Z for low ass velocity (ass velocity < 650 kg/s- ) profiles. Data points (+), i.e. K versus Z values, in the figure were obtained by iniizing the pseudo-variance. Z is defined by Equation (9) with α = , β = , γ = , and ω = 0.1. The solid line denotes an analytical fit to the K-Z data. As for the high ass velocity cases, the downhole pressure profiles for all the low ass velocity cases in the dataset (Table ) were siulated using the correlation for K (solid line, Figure 5). The latter coputed profiles are for the ost part in good agreeent with (1) easureents, and () calculated profile (best-atch) using an adjustable holdup correlation (Section ). 5. COMPARISON WITH SPINNER DATA The rotational speed of a spinner is proportional to the ass-averaged velocity of the fluid ixture passing through the spinner. Except for annular flow, the ass-averaged velocity indicated by the spinner should correspond closely 5

6 Table : Coparison of wellhead pressures for low ass velocity profiles (ass velocity < 650 kg/s- ) coputed using the correlation for K(Z) shown in Figure 5 with data Discharge Profile A-4 Well P-data P-cop δp ** A C C C C C B B B KE KE KE1-17b KE1-a S CS ** δp = P-data P-cop Total (15 profiles) 0.04 to the ass-averaged velocity of the entire flow strea (Gang, et al., 1990). The spinner response f is related to ass-averaged fluid velocity v and cable speed v c as follows: f = v ( v) + c (11) where and c are the calibration constants. The ass-averaged ixture velocity v can be coputed fro the in situ liquid and vapor saturations, densities, and velocities. v ( ρlslvl+ ρgsgvg) = ( ρ S + ρ S ) c l l g g (1) In geotheral wells, the liquid (i.e. water) density is uch greater than the vapor (i.e. stea) density; this eans that the ass-averaged ixture velocity is a strong function of the in situ liquid volue fraction S l. Thus atching the spinner data can provide an independent check on the holdup correlations developed in Sections 3 and 4. Coparison of spinner data with the coputed fluid velocity requires a certain aount of care. Spinner data are often quite noisy in two-phase flow, and ust be soothed for a eaningful coparison. In addition, spinner surveys often display anoalous response (e.g. a decrease or no change in rotation rate as the spinner tool is traversed up the hole, or a less than coensurate change in rotation rate at a discontinuous change in well diaeter) due to iproper centering of the tool in the well or a possible teporary flow obstruction in the spinner tool. Obviously, such spinner data ust be discarded. The soothed spinner data for well A-4 are copared with the coputed spinner response in Figure 6. The spinner data for well A-4 (Figure 6) exhibit an anoalous response Vertical Depth, eters Spinner, rev/s Figure 6: Coparison of the soothed spinner response (triangles) with the coputed spinner response (dashed line) for well A-4. (decrease/no change in the rotation rate) in the depth range As can be seen fro Figure 6, except for the depth range 0 150, the soothed spinner response shows a good agreeent with the coputed curve. Siilar results for all the other wells with spinner data are given in Garg, et al. (004). Barring very few exceptions; the spinner easureents are in good agreeent with the coputed response (Garg, et al., 004). Thus, taken as a whole, the spinner easureents tend to validate the holdup correlations developed in Sections 3 and CONCLUDING REMARKS The principal goal of the present work is to use high quality data fro flowing geotheral wells to devise new liquid holdup correlations for geotheral applications. To ake the proble tractable, it was decided to at first develop a holdup correlation for only the cased section of geotheral wells. The holdup correlation presented in the preceding sections displays considerable proise for siulating twophase flow in the cased section of geotheral wells. Future plans call for the forulation of a holdup correlation for the open hole/slotted liner section of geotheral wells. ACKNOWLEDGEMENTS This anuscript has been authored by a contractor of the U.S. governent under DOE Contract DE-AC07-99ID1377. Accordingly, the U.S. Governent retains a nonexclusive, royalty-free license to publish or reproduce the published for of this contribution, or allow others to do so, for U.S. Governent purposes. We also wish to express our thanks to Unocal Corporation and Caithness Operating Copany LLC for aking their proprietary well data available for this study. REFERENCES Ansari, A. M., Sylvester, N. D., Sarica, C., Shoha, O., and Brill, J. P.: A Coprehensive Mechanistic Model for Upward Two-Phase Flow in Wellbores, SPE Production & Facilities, , May (1994). Aziz, K., Govier, G., and Fogarasi, M.: Pressure Drop in Wells Producing Oil and Gas, Journal Canadian Petroleu Technology 4, 38 48,

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