Mathematical Models of the Fluid Flowing for Geothermal and Hydrocarbon Wells

Transcription

1 Proceedings World Geotheral Congress 005 Antalya, Turkey, 4-9 April 005 atheatical odels of the Fluid Flowing for Geotheral and Hydrocarbon Wells Olga B. Vereina Geological Institute of Russian Acadey of Science, Pyzhevsky per., 7, oscow 907, Russia Keywords: geotheral well, gas-condensate well, HOLAsiulator, two-phase flowing, itical flow rate ABSTRACT Estiation of production response of the geotheral or hydrocarbon reservoir includes odeling i) the processes in reservoir during exploitation and ii) fluid flowing in the production and re-injection wells. Therefore, separate atheatical odels for siulation of the conditions in reservoir and flowing in the wells are to be jointed as one concept. The paper presents the coparative analysis of odels for hydrocarbon and geotheral fluid flowing in the well. Analogy between two types of two-phase fluid (gas condensate and stea water) is drawn and differences are discussed. Soe calculation results for wells of Karachaganak gas-condensate field and utnovsky geotheral field are presented.. INTRODUCTION The odel for calculating the ain paraeters of flowing in the wells (production flow rate, enthalpy, coposition of extracted fluid) is necessary in developent and exploitation of both geotheral and hydrocarbon fields. It is possible to draw analogy between flowing in the well for hydrocarbon (gas condensate) and geotheral (stea water) two-phase systes (Vereina, 00). In both cases two-phase fluid of ulti-coponent coposition is under consideration and atheatical odel includes the sae basic equations followed fro conservation laws for ass and energy. In the sae tie, the following differences should be taken into account:. Geotheral syste is non-isotheral while gascondensate flowing ay be assued isotheral to siplify calculations.. In the case of gas-condensate well, it is possible to solve a proble assuing alost constant gas content, but for geotheral well water-stea phase transitions and stea content variations along the well are significant. The share of stea in geotheral fluid ay vary in the range fro 0 to (e.g., when water-saturated sea is uncovered by a well, pressure drop can cause boiling along the well that will produce pure stea). 3. Closure equations are different because they desibe the properties of certain fluid. The present work considers several odels. These are HOLA-siulator designed for odeling ulti-feeone geotheral well, and two odels, non-isotheral two-rate and siplified (isotheral hoogeneous), designed for estiating paraeters of itical flowing for hydrocarbon wells. Soe illustrating results for utnovsky geotheral field and Karachaganak gas-condensate field are also presented.. WELLBORE SIULATOR HOLA AND ITS USE FOR PLOTTING PT-PROFILES.. Brief desiption and governing equations The ulti-feeone geotheral wellbore siulator HOLA developed by Griur Bjornsson and Thordur Arason (Bjornsson et al., 993) allows one to reproduce the teperature and pressure profiles in flowing wells and deterine the contribution of each feeone for given discharge conditions. The code is written in Fortran prograing language and executable The siulator HOLA uses the following basic assuptions. The flow within the well is assued steady-state, since changing reservoir pressures are allowed. The siulator can be used for single and two-phase flows in vertical pipes, and calculates the flowing teperature and pressure profiles in a well. The properties of stea and water are calculated according to forulae presented by the International Forulation Coittee in 967. Two sets of equations are used to represent the flow of fluid in a geotheral well. The flow in the well, between the feeones, is represented by one-diensional steady-state oentu, energy and ass balance equations. When a feeone is encountered, the equations of ass and energy balance between the fluid in the well and the feeone are used. To solve these equations fully defined boundary conditions (wellbore geoetry, lateral ass and heat flow) and flow conditions at one end of the syste (inlet conditions) are required. The governing equations are solved by nueric ethods in sall, finite steps along the well. If a feeone is encountered, the known paraeters of inflow, or outflow (ass and energy), are used to continue calculations. The equations of ass, oentu and energy flux in a vertical well are written as: d 0 () dp dp dp dp () fri acc pot de t ± Q 0 (3) The plus and inus signs correspond to downflow and upflow, respectively. The pressure gradient includes three ters: wall friction, acceleration of fluid and change in gravitational load over. The equation of interaction between the well and the reservoir is:

2 feed PI feed k r kr + ( pr pw ), µ µ (4) where relative pereabilities are calculated by linear relationships: k rg s; k rl -s (5).. Plotting pressure and teperature profiles for utnovsky geotheral field The wellbore siulator HOLA offers six odes of calculating downhole conditions in geotheral wells. Figure 3: The pressure and teperature profiles in the well 4 located at the Dachny site of utnovsky geotheral field, plotted by HOLA-siulator. The pressure and teperature profiles are represented for soe wells located at the Dachny site of utnovsky geotheral field at Kachatka. Because of intensive developent of the field the odeling of its response to exploitation is of great iportance. This odeling includes the estiation of conditions in the wells, which is to be joined to coplex ulti-paraeter siulator like TOUGH. Figure : Location of wells at the utnovsky geotheral field (altseva et al., 00). Table Inpute paraeters for wells 0 and 4 (Assaulov and Assaulova, 000) Well Depth () Flowrate (kg/s) Wellhead enthalpy (kj/kg) Wellhead pressure (bar-a) The location of the wells of utnovsky field is shown in Figure, and the results of plotting pressure and teperature profiles for wells 0 and 4 are presented in Figures, 3. The input data is shown in table. For each well one feeone is proposed and well depth is considered until feeone. The breaking point of teperature curve for well 4 corresponds to phase change; below this point the fluid in the well is in single-phase (liquid) state. In the well 0 the fluid is in two-phase condition; dryness is equal to 8.7% at the wellhead and deeases with depth alost uniforly. The paraeters of feeones for wells 4 and 0 are shown in table. Figure : The pressure and teperature profiles in the well 0 located at the Dachny site of utnovsky geotheral field, plotted by HOLA-siulator. The current paper considers the first ode of HOLAsiulator, the case of Outlet conditions known at the wellhead (Bjornsson et al., 993). In this case siulator reproduces pressure and teperature profiles fro given wellhead conditions and given flow rates and enthalpies at each feeone (except the botto one). Table Feeone paraeters for wells 0 and 4 calculated by HOLA-siulator Well Enthalpy Pressure (kj/kg) (bar-a)

3 3. ODELS FOR CALCULATING THE CRITICAL FLOWING PARAETERS In the developent and exploitation of underground fluids itical well flowing can occur, which ay result in heavy technical and ecological consequences. It is especially actual for deep-seated hydrocarbon fields of ulticoponent coposition with a great content of H S and CO as well as for high-teperature hydrotheral systes, since their exploitation ay be accopanied by phase transforations both in a sea and in a borehole. Estiations of itical flow rate, wellhead and well-botto pressure are necessary for choice of the anner of blowout daping and require special calculation odel. The follows are odels designed for calculating itical flow rates of gas-condensate wells. 3.. Non-isotheral heterogeneous (two-rate) odel for gas-condensate well In (Basniyev et al., 99) the following one-diensional two-rate odel for calculating the itical flow rates of gascondensate wells is presented. This odel involves following equations of stationary two-phase ulticoponent flow in the vertical well under exploitation: ass conservation equation: + const (6) oentu change equation: dp g λ d DS ( ϕ) ϕ S ( ϕ) ϕ energy conservation equation: d w w i i g πdk ( T T ) The following equations are need to enclose the equations set (6) (8): state equations for liquid and gas (stea) phases: R (7) (8) g g (P,T) l l (P,T) (9) epirical forulae for real gas (stea) content and fluid hydraulic resistance coefficient which are different at various fluid flow regies (Basniyev et al., 99): µ ϕ ϕ β,,, Fr, Re, We (0) µ µ λ λ ϕ,,, ε, Fr, Re, We () µ forulae for Reynolds, Froud, and Weber nubers, and relations resulting fro definitions of two-phase flow paraeters. Teperature distribution (initial teperature) in reservoir is defined by expression: T R T nl + G (z - z nl ) () Interaction between productive bed and well is desibed by the two-ter inflow equation: P k - P з Аµ ср *z ср c + Bz ср c P R - P b AQ + BQ (3) (3a) A and B designate filtration resistance coefficients defined by well testing under stationary regies (Basniyev et al., 99). The following additional functions are introduced for eating the algorith: Ф (/S )[ g /ϕ g + l /(-ϕ) l ] Ф 3 g (i g +w g /)+ l (i l +w l /) Ф (λ с /D)Ф +g c Ф 4 g c -πdk(t 0 -T) (4) After substituting (4) into equations (7) - (8) we have: where dp/ (/ ) [Ф 4 Ф / T - Ф Ф 3 / T] (5) dt/ -(/ ) [Ф 4 ( + Ф / p) - Ф Ф 3 / T] (6) Ф / T Ф 3 / p - ( + Ф / p) Ф 3 / T (7) Set of equations (5)-(6), (6), (9)-() at given boundary conditions (botto-hole pressure and teperature) allows us to calculate distribution of pressure, teperature and other flow characteristics in the wellbore. Reservoir properties are considered by equation (3) allowing us to define botto-hole pressure depending on ass flow rate. Critical flow rate is defined fro pressure and teperature jup on the wellhead. atheatically it eans that pressure and teperature derivatives at wellhead convert to infinity: dp/ dt/ (8) This requires denoinator in (5) - (6) (or expression (7)) to be equal to zero: 0 (9) Wellhead flow rate defined by (9) is called itical flow rate (or, free flow rate ). It is axial possible flow rate of the well. 3.. Isotheral hoogeneous odel for gas-condensate well The paper (Polyak (Vereina), 996) presented calculating itical paraeters (flow rate and wellhead pressure) for gas-condensate wells using a sipler, isotheral hoogeneous hydrodynaic odel proposed in (Astrakhan and Rozenberg, 988). The use of such siplified odel is justified because the overwheling ajority of itical blowouts occurs during the field prospecting (at the stage of the drilling and inflow testing) and at initial stages of the field exploitation. Under these conditions, all necessary data on structure of the field, fluid properties and coposition, features of fluid flowing within a bore hole and phase transforations (gas-liquid) are unknown beforehand, so that use of coplicated hydroechanical ethods for the desiption of itical well flowing is inexpedient. 3

4 The siplified odel considers joint flowing in reservoirwell syste and based on following assuptions: flowing is state and one-diensional, hoogeneous (single-rate), and isotheral. oreover, the liquid phase is assued incopressible and gas phase is desibed by Clapeyron equation. Two-ter filtration equation is assued, and the flowing direction (i.e. upflow) is assued as positive. According to these assuptions pressure gradient for onediensional hoogeneous flowing can be forulated as: where dp xrt c S λ D S x x + + c P g x + ( x) (0) () Condition of itical flowing is conversion of denoinator in (0) into zero at z0. After substitution the Clapeyron forula: P () RT and taking into account expression (), equation (0) for itical flowing is written as: λ x g p dp D + p xrt + xrt/p + p p ( x) / (3) Botto-hole pressure is defined fro equation (3a) desibing filtration in reservoir. Critical flow rate is defined as: Q (4) at Using expressions () and (4), the equation (3) can be written as: where P ** ** R Pb A P + B P (5) A ** at AS xrt BS xrt ** B at (6) Set of equations (3) and (5) define wellhead and well botto pressure at itical flowing. Critical flow rate is defined fro equation, derived fro () and (4): Q S p (7) xrt at The paper (Polyak (Vereina), 996) also offers odification of isotheral hoogeneous odel for the case of variable sectional area of the well. Boreholes ay be distorted by contracting and pulling stresses, which are acting in rocks; thus, the sectional area of the well becoes unequal in 4 various depth intervals. These fluctuations are to be considered in odel for itical paraeters calculation. Pb, Pa Hole , 0, 4 0, 6 0, 8 P, Pa Figure 4: Calculation of itical paraeters for well 04 of Karachaganak gas-condensate field (Polyak (Vereina), 996). Pb, МПа Hole , 0, 4 0, 6 0, 8 P, Pa Figure 5: Calculation of itical paraeters for well 05 of Karachaganak gas-condensate field (Polyak (Vereina), 996) Critical flow rate calculation: coparing the results for different odels Two variants of calculation are presented in (Polyak (Vereina), 996):. If gas content is considered to be constant, then equation (3) is solved analytically. In this case its solution is written as: where: Ф Ф(p з,p кр ) - Ф(p кр,p кр ) H (8) ( p, p ) kp λ D p kp p kp dp p - x + + p xrt l xrt/p g + ( x )/ So, wellhead and well botto itical pressure are defined fro equations (5) and (8). Critical flow rate is defined fro equation (7).. If gas content depends on pressure, equation (3) is solved by nueric ethods, and graph is plotted for bottohole pressure depend on wellhead itical pressure. Another graph is plotted according to equation (5), and oss-section point is to be found. This point corresponds to solution (see Figs. 4, 5). Critical flow rate is defined by the sae way as in previous case. l

5 In (Basniyev et al., 99) and (Polyak (Vereina), 996) calculations of itical paraeters were perfored for the wells of Karachaganak gas-condensate field (located to the north of Kaspian Sea). Coparison of results for different odels has shown that the siplified (isotheral hoogeneous) odel can be used to estiate free flow rates for the wells under consideration. axiu disepancy didn t exceed 6 %. It ay be explained by validity of siplified odel at enough high both gas content and flow rate of the fluid in the wells, which is typical for the given field. However, the siplified odel is inadequate for calculating itical wellhead pressure, since it results in extreely overstated estiations (the axiu difference between values obtained fro coplicated and sipler odel reaches 55 %). Nevertheless, for singlephase fluid (pure gas) the estiations both itical flow rate and itical wellhead pressure using the siplified odel well agree with the results obtained using ore coplex odel (the axial disepancy is of and 3 %, respectively). 4. CONCLUSIONS The developent and exploitation of geotheral and gascondensate fields need the odel for calculating the ain paraeters of flowing in the productive and re-injection wells. The current paper considers existent odels for desiption of well flowing in both cases of fluid and soe calculation results are presented. The following inferences can be ade:. Considering the atheatical odels for the flowing in geotheral and hydrocarbon wells we can see that they are siilar. Both the include the basic equations of ass, oentu and energy balance for flowing in the well and equations of inflow (or outflow), i.e. equations of wellreservoir interaction. The ain difference is the desiption of fluid properties. oreover, unlike the flow in geotheral well, the flow in gas-condensate well ay be considered isotheral and gas phase content ay be assued constant.. The ulti-feeone geotheral wellbore siulator HOLA is considered, and soe results of plotting pressure and teperature profiles are presented. Such odel desibing the flow in wellbore is necessary for odeling the field response to exploitation, and is to be joined with coplex ulti-paraeter siulator, like TOUGH. 3. Two odels for calculating the itical flow paraeters for hydrocarbon wells, non-isotheral two-rate and isotheral hoogeneous, developed in previous works, are exained as well as coparison of their results is presented. It was shown that siplified (isotheral single rate) odel allows us to predict itical flow rates for hydrocarbon wells. 4. Critical flowing ay occur during exploitation of geotheral fields too, but this specific case need the special calculation odel. Acknowledgeents. y deepest thanks to professors of Gubkin Russian State Oil-Gas University, Dr. I.. Astrakhan, Dr. V.I. Isayev, and Dr. D.G. Polonskii for their guidance and providing the data on Karachaganak field, to Dr. Gudni Axelsson and Dr. Griur Bjornsson, for providing software package ICEBOX and instructions on using HOLA-siulator, and to Dr. Oleg Povarov for providing the data on utnovsky field. NOENCLATURE A, B filtration coefficients A *, B * reduced filtration coefficients D hole diaeter E energy flux G geotheral gradient g acceleration of gravity H well depth i enthalpy K heat transfer coefficient ass flow rate P pressure PI productivity index Q flow rate R gas constant S well sectional area, s voluetric stea saturation T teperature w average velocity x ass gas (stea) content z vertical coordinate Greek letters β voluetric gas (stea) content λ hydraulic resistance fluid density ϕ real gas (stea) content Subsipt 0 standard conditions gas (stea) phase liquid (water) phase at atospheric conditions b well botto itical feed feeone h wellhead ixture (two-phase fluid) nl neutral layer r reservoir w well REFERENCES Assaulov, S.G., and Assaulova, N.P.: utnovsky geotheral field DATABASE. In: Drilling and reservoir developent. utnovsky Independent Power Project, internal docuentation. (000) Astrakhan, I.., and Rozenberg, G. D. Definition of liiting flow rate of the flowing well for two-phase fluid (In Russian). In: Filtration of non-unifor fluids. oscow: VNIIGAZ, (988) Basniyev, K.S., Rozenberg, G.D., Isayev, V. I. et al. Thero-hydrodynaic odel for calculating free flow rate of gas-condensate well, inforation review (In Russian). oscow, VNIIEGAZPRO, 36 pp. (99) Bjornsson, G., Arason, T., and Bodvarsson, G.S.: The Wellbore Siulator HOLA, version 3.. User s Guide: Orkustofnun, Reykjavik. (993) altseva, K.I., Assaulova, N.P., Kozlov, A.E., et al.: Project of pilot and coercial developent of Dachny site of utnovsky geotheral field, internal report (in Russian). SC Geother, Akva Ltd., Elizovo, Kachatsky region. (00) Polyak (Vereina), O.B. Coparative analysis of various odels of ulti-phase edia to calculate itical flow 5

6 rate of gas condensate wells: r. Th. (In Russian) Gubkin Russian State Oil and Gas Acadey. (996) Vereina, O.B. Evaluation of itical flowing paraeters of productive wells (In Russian). In: The Earth s Theral Field and Related Research ethods: Col. Res. Articles. oscow, RPFU Publ., (000) Vereina, O.B. An Estiation of Paraeters of Geotheral and Hydrocarbon Productive Wells. In: The Earth s Theral Field and Related Research ethods: Proceed. of International. Conf. oscow, Russia, June 7-0, (00) 6