MOL Budapest HUNGARY 1311

Transcription

1 MODELING of TEMPERATURE CONDITIONS of REINJECTION in GEOTHERMAL RESERVOIR'S MOL Budapest HUNGARY 1311 Key words: rmal simulator, reinjection, Hungary ABSTRACT A rmal simulator is presented that originally was developed to model multiphase fluid flow in hydrocarbon reservoirs. The model has a modular structure. The module, and fluid module are highly independent parts of model. The integrated finiteform is used. changing fluid module ( multicomponent, multiphase hydrocarbon module) to a one component, steam--water system, model can be used to simulate geormal A new numerical solution technique is applied: pressure and temperature distributions are calculated directly, without iterations from mass and energy balance equations. No mass and energy balance errors occur. model used to decreasr rates of Transdanubian spa, where water supply is coming from temperature sources a or application simulated pressure and temperamre distributions gcormal field Computer simulators of geormal reservoirs have been used since numerical techniques applied by geormal models are very similar to those of reservoir simulators as equations to he solved are essentially same. There are, however. een above two type of reservoirs simulation problems) coming from se two sources. is worth to mention here great and conditions of geoimal reservoirs. and systems of hydrocarbon hen enhanced oil technologies. Fractured, host rock can be found in both groups ofcases. effects of faults can more dominant lor geormal reservoirs. To simulate hydrocarbon displacement twothree-phase. multicomponent reservoir simulators have been used since mid sixties. while solution of rmal models steam injection and in-situ combustion) have been published since first two decades of model models used for different (such as hvo- or three-dimensional flow of black oil or compositional fluid description). Since early eighties it is a usual to apply so called general purpose simulators. mamatical backgrounds of models are nonlinear and equations. Darcy's equation to momentum equation. and equations describe behavior system. It can be seen in technical literature that number of solution techniques is great enough due to many variables of' this complex system. Among ors. numerical solution techniques depend on selection of calculated from conservation equations. A natural selection of variables characterization multicomponent hydrocarbon fluid systems could be by applying a pseudo gas, and a pseudo oil component. This simplification, with empirical properties, (such as relative permeabilities, depending on phase saturations) have led to black oil models. Then, phase saturations are used as primary variables, instead of overall component densities. The temperature has been typical pnmary variable rmal hydrocarbon displacement models, too. These latter variables are not suitable to describe all states of a fluid system. The generally used IMPES (implicit pressure, explicit saturation) method has made it difficult to to cotnpositional modeling in seventies. The pressure is primary variable of models. The enthalpy is used as primary variable of geormal models. This was applied, and Mercer (1976). These variables are suitable in cases of one- or two water phases. as well. In models of hydrocarbon recovery were published using primary variables instead of temperature et too. building a simulator. an appropriate numerical technique can help to write a versatile. and concise program code, to separate parts a simulator, such as fluid characterization, geometrical parameters of' model, time-discretiation techniques. In this a solution presented that is suitable to serve as a basis of a general of this was simulate isormal hydrocarbon displacement (Acs al. and later extended lor (Farkas and rmal model was applied to simulate some geoimal problems. equations of model describe consewation of components. and total internal energy. us use notation lor overall component inass density NP = and j-1 for total specific internal NP = + - In 1 and notations: is is of phase is phase is mass concentration of component in phase and u is subscript r denotes rock. mass and energy equations can be written as : 1985

2 where is = At - + At = NC change of internal energy in a volume element is following : In 3 5 notations: p is pressure, is mass rate of component i, is specific enthalpy of phase j, is conductivity coefficient, is temperature, is source rate, is energy loss rate, K is absolute permeability, k, is relative permeability, is viscosity, g is gravitational constant, and z is depth. An additional equation, in-situ "volume balance" equation is added to complete basic balance equations. The original volume balance equation was given for isormal systems by Acs et al. (1985). Now, this equation is modified to take account rmal effects, too: At - At The second term of right-hand side of Eq. 12 is explicitly calculated part of energy change. The heat conduction and heat loss are considered explicitly. The production term is discussed elsewhere 1993). Now. pressure equation will be derived from volume balance equation Eq. 6. The first order Taylor approximation is applied to express equality of volumes at new time level. here subscript and p refer to fluid, and pore respectively, is total internal energy, m is component mass vector. The volume balance equation expresses, that fluid system should pore system during displacement process. THE EQUATIONS The integrated form of 3 and 4 are used = and = The difference equations will be written for changes of primary variables where, superscripts n, and o denote new and old time level. mass rates I, (unit pressure difference), and (unit time) are introduced We can substitute difference approximation of energy change into third term on left-hand side of Eq. 13, and difference approximations of mass changes into last term on left-hand side of Eq. 13. By changing order of summation for components and neighboring elements, rearranging equation, and using notation for total compressibility = - and and denote volume rates, av = = concise form of pressure equation is is mass rate of component flowing through k-th boundary of volume element (k-1. is explicitly calculated mass rate of volume element denoted by subscript Similarly, enthalpy rates (unit pressure difference) and (unit time) To show influence of rmal we compare 16 with isormal pressure equation of Acs et al. which is following: + - Now, it can be observed, that Eq. 16 differs fiom Eq.17 in volume rates. We can see clearly now, how rmal effects isormal volume rates. will be used in difference equations, too. In 9 and upstream are used. Then, changes can be written as The governing difference equations are Eq. 12 and Eq. 16. set of equations makes it possible to apply a sequential solution technique for primary variables, for pressure, masses and total internal energy. First pressure changes are determined implicitly from 16 written for all volume elements. Then, changes can be calculated explicitly fiom Eqs. 11 by volume elements, finally change of total internal energy can be derived explicitly from Eq. 12 by volume elements. 1986

3 After determining new primary variables, new is calculated. The variable T is function of and m. The inverse of function is applied. The change of intemal energy can be written from change of function too We have already determined changes of primary variables Am,, and in last timestep, and know partials (referring to beginning of timestep). temperature change can be expressed from Eq. 18. At saturation temperature, AT cannot be expressed from 18, because n, is not with respect At steam saturation temperature, however, temperature can simply be calculated from saturation pressure 5. Calculation of total intemal energy changes (Eq. 12). 6. Calculation of new temperature (Eqs. 7. Calculation of all fluid properties from new set of primary variables Appendix of 1993). Important features of this method: primary variables of model can apply to any arbitrary fluid system, grid calculations, and fluid--rock characterization are totally decoupled from solution technique of differential equations, - all coefficients of pressure equation have physical meanings. The above method only pressure implicitly, masses and energy are calculated explicitly. A fully implicit version can also be realized. An isormal extension of this volume balance technique is discussed by Farkas (1993). The implicit energy calculation has not yet been worked out. PRESSURE AND TEMPERATURE CORRECTIONS The term in Eq. 16 expresses volumetric error committed in earlier timesteps. error is originated from explicitly calculated volume derivatives. volumetric error proved to be an appropriate control of timesteps. The timesteps are chosen in such a way that volumetric error be kept under a negligible volume. This volume discrepancy will be corrected in next timestep by modifying pressure equation. Consequently, this correction term cannot be applied for incompressible systems, and should be small enough of pore volume), orwise it can dominate pressure equation. This dominance occurs especially at small timesteps. Now, a new correction step has been introduced that proved to be very useful in rmal model presented here. The aim of this step is to get correct temperature. After determining new temperature from Eq. 18, we can calculate all rmodynamic properties at new time level, including fluid volume and total energy of volume element. Because of first-order approximation errors, however, total internal energy calculated from function may not be equal to internal energy calculated from energy balance equation Eq. temperature correction step helps to get equality by appropriately T. The Newton, or or iteration method can be used to find T from nonlinear Eq. initial value of T, calculated from Eq. 18. is a good estimation of new temperature. In our practice. generally, one iteration step is enough to reduce under relative energy error STEPS OF THE The calculations of a timestep consist of following 1. Determination of mass and enthalpy rates:, for each boundary surface (Eqs. 9-10). 2. Determination of compressibility and volume and of pressure equations (Eqs ). 3. Solution of system of linear pressure equations (Eq. for all volume elements). 4. Calculation of component changes 11 APPLICATIONS The model was applied to simulate geormal reservoirs. A simulation study examined cause of temperature decrease of Transdanubian Heviz. spa, where water supply is coming from different temperature springs of a huge karstic fractured reservoir. In this study, appropriate handling of faults caused difficulties, especially under rmal lake that was formed above intersection of two faults. Heviz is situated near to lake Balaton (see Fig. In eighties. a great amount of water (about millions had to be produced from extended Transdanubian karstic reservoir to ensure bauxite mining activity at about 25 km north east from at Nyirad. The amount of water produced from rmal wells at Heviz area was increasing in this period, too. At end of seventies natural water to lake was 450 At end of eighties rate decreased to about 300 purpose of simulation study was to examine cause of reduction of natural water source rates. The simulation study examined only a small km x 2.1 km) area of huge reservoir (Fig. 1). We had detailed pressure and temperature distributions of this area due to many observation wells, and measurements in regions. The two permeable layers of area are separated with an impermeable layer (Fig. The layers at north east comer of simulated area are direct connection with surface (see Fig. 2). The temperature and pressure distributions (in 1979 and in 1988) can be seen on Fig A special grid was constructed applying radial elements around rmal water wells of area, and to model lake Fracture volume elements were applied to calculate water flow through fractures. The connections of two permeable layers were assumed to exist mainly through fracture elements. The water could flow to only through fracture elements of two main fractures of area. The areal grid points of both permeable layers (not including radial, and fracture grid points) are demonstrated in Fig. fluid model assumed to have two components (water and air) to consider free surface volume elements. The measured pressures were given as boundary conditions of model. For water column pressure (at a depth of 40 m) was prescribed. For rmal wells, water production rates were given. To complete conditions, average water influxes of free surface elements were assumed too. The parameters of fracture elements were determined by applying total water rate of springs hnown at 1979, ( maximum error of simulated water level was 1.5 m, see Fig. 4). When using matched fault parameters, water outflow to lake was simulated an error of by time of Then, water level derivation fiom measured values was about one meter (see thin lines in Fig. When omitting production wells, natural 1987

4 Areal grid points (without radial and fault elements), connections of volume elements, and main faults Figure 1. A map of Heviz surroundings with Nyirad, and model size dlstance. m Figure 2. A simplified SW-NE cross section Figure 3. Temperature distribution in Pannon thln show of measured and values Figure 5. Measured water levels in Pannon layer (m on Balti see level), 1988 thin lines show differences of measured and simulated values 1988

5 outflow rate to lake was increased only by of total production of rmal wells. calculated temperature of water flowing to lake was 9 colder than measured average of sources. mining activity, water production at had been stopped at early nighties hecause of environmental damages. Thus, to clear above questions. and examine or simulation problems no additional support was given. 'The above work was done in Since stopping water production at Nyirad, a rise of natural springs' rates in can he observed. Now. this rate is over 370 In an or application model was used to simulate geormal field of Szentes that is a part of huge extent Pannon sandstone layers. nnal well was drilled in sixties to heat greenhouses. Belonging to number of wells had to forty by end of eighties, while average value of well rates was continuously decreasing from 600,000 to 300,000 year. A two dimensional niodel was used to of production including only a part (about km 100 km area) of Pannon sandstone problem was to give appropriate water rates refemng to boundaries model to connection between whole and model areas. boundary conditions were used including two limits: (1) closed reservoir, (2) unlimited water and to assume pressure declines at of area. Applying variations of boundary conditions. simulation runs showed no of average pressure level in near area first five years of production. The pressure level of geormal field depended on local production individual wells. and long distance boundary conditions ensuring water supply of model. pressure level could partly be regenerated during summer period when production was reduced or stopped, but water supply rate of area was less than produced water rate. of water injection was examined. The injection wells were simulated about from production wells. distance enough to cold water and pressure decline could be partly reduced. examine behavior 01 model in of two phases. a steam injection problcm (an SPE comparative solution project) was simulated (see 1993). thanks to for and preparation all reservoir, and production data used in model. S.. and Farkas, (1985) General Purpose Compositional Model, Pope. and (1991) Development a Consistent. Fully Implicit; Equation of State. Simulator, paper presented at Eleventh Symposium on Simulation Adaptive Implicit Volume Balance paper SPE 25973, 12th Symposium on Simulation, Farkas, P. (1993) A Direct Balance Technique for Implicit Steam models. paper SPE Advanced Series, 1994 Faust. and (1976): Analysis of Difference and Finite-Element for paper presented at Fourth Symposium on Reservoir Simulation Watts. and Fully Fully Reservoir Simulator for and Or Complex Reservoir Processes", paper 1252 presented at I th Symposium on Simulation 1989