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Recent developments in numerical simulation techniques of thermal recovery processes
Journal of Petroleum Science and Engineering 26 Ž2000. 283–289 www.elsevier.nlrlocaterjpetscieng
Recent developments in numerical simulation techniques of thermal recovery processes M. Tamim a , J.H. Abou-Kassem b,) , S.M. Farouq Ali c b
a Bangladesh UniÕersity of Engineering and Technology, Bangladesh Chemical and Petroleum Engineering Department, UAE UniÕersity, Al-Ain 17555, United Arab Emirates c UniÕersity of Alberta, Alberta, Canada
Received 12 December 1998; accepted 15 December 1999
Abstract Numerical simulation of thermal processes Žsteam flooding, steam stimulation, SAGD, in-situ combustion, electrical heating, etc.. is an integral part of a thermal project design. The general tendency in the last 10 years has been to use commercial simulators. During the last decade, only a few new models have been reported in the literature. More work has been done to modify and refine solutions to existing problems to improve the efficiency of simulators. The paper discusses some of the recent developments in simulation techniques of thermal processes such as grid refinement, grid orientation, effect of temperature on relative permeability, mathematical models, and solution methods. The various aspects of simulation discussed here promote better understanding of the problems encountered in the simulation of thermal processes and will be of value to both simulator users and developers. q 2000 Elsevier Science B.V. All rights reserved. Keywords: reservoir simulation; modeling; thermal recovery; developments
1. Introduction Reservoir simulation has gained confidence as evidenced by the frequent use of simulation results in the reservoir engineering decision-making process. Thermal simulators started evolving in the early 1970s. Thermal computational modeling is difficult due to many complex mechanisms including high degree of nonlinearity. Better physical understanding of the mechanisms involved in thermal operations and the advancement of implicit computational tech-
)
Corresponding author.
nique helped thermal simulators to gain maturity. These improvements and successful applications have given the petroleum and software engineers the confidence to develop commercial simulators. This resulted in the transition from command-based formatted data input to colorful graphical user interface. Therefore, the petroleum industry and researchers have moved more towards using commercially available simulators in the last decade. Recent studies are concerned with improving the problem areas of the existing simulation models. Some of these topics and one new modeling approach in thermal numerical simulation technique are discussed in this paper.
0920-4105r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 4 1 0 5 Ž 0 0 . 0 0 0 4 2 - 5
284
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
2. Grid refinement In a thermal process, simulation results are largely influenced by improper modeling of near-wellbore phenomenon and handling of localities where variables change rapidly. Therefore, it is possible to predict inaccurate steam injection pressure or steam override behavior at the wellbore. Coning behavior in thermal operations could also be erroneously predicted due to improper modeling. Local grid refinement is a natural process to handle these situations. In grid refinement procedures, it is common to find irregular connections that evolve from the coupling of coarse and fine grid blocks. This produces matrices with complex structure. Rodrigues and Dickstein Ž1996. addressed this problem using a composite mesh based on a finite volume technique. The grid blocks were distributed among four groups. The first group comprises those blocks in the basic coarse Žparent. mesh that have no connections with refined blocks. The second group comprises those blocks in the refined mesh that have no interface with the basic coarse mesh. These two groups have the usual seven-point operator with additional balance equation for each of the intermediate time steps. The refined blocks at the interface were put in the third group. A refined block ijk and the parent coarse block IJK would have the interface as i q 1r2, j,k. The accumulation, source and flux terms for the other faces remained the same. Pressure at t nq m r M at point x Iq1, jk is required to calculate the pressure gradient at the interface. This point is not in the mesh ŽFig. 1.. Interpolation in time and space would be required. The simplest of the interpolation schemes was used; the interpolation by a constant nq m r M nq 1 pIq1, jk s pIq1, JK
Ž 1.
Boundary coarse blocks were put as the fourth group. Balancing the sum of the fluxes of the refined blocks with their parent coarse block for all intermediate time steps. Their multigrid procedure to solve the set of nonlinear equations allowed for local time stepping. Apart from the abovementioned discretization method ŽCartesian hybrid grid., other methods of discretization such as control volume finite element ŽCVFE. grids and hybrid CVFE grids have also been used to address near-wellbore phenomenon. Rapidly
Fig. 1. Coarse and fine grid interface Žpressure interpolation required at x ŽRodrigues and Dickstein, 1996..
changing variables like temperature, pressure, or saturation require smaller grid blocks mostly around the wellbore. Hiebert et al. Ž1993. found that Cartesian and CVFE hybrid grids have the capacity to accurately model near-well phenomenon. CVFE grid has more flexibility and may provide easier coupling with the cylindrical grid that is used for a more accurate flow modeling around the wellbore. In recent years, improved drilling techniques coupled with lower cost and efficient use have made horizontal drilling more popular in many thermal operations. Integration of the traditional sinkrsource modeling of these wells with a field-wide thermal simulator neglects compositional, energy, and frictional pressure variation along the wellbore. In a discretized wellbore model, each section of the wellbore is treated as a grid block that handles all property variations in a unified manner with the reservoir. Peaceman Ž1983. and Babu and Odeh Ž1989. reported equations for well geometric factors for blocks penetrated by horizontal wells. Oballa et al. Ž1997. explored the advantages and limitations of discretized wellbore models. They recommended that caution be exercised when using the more complex-discretized-wellbore model. Numerical difficulties may arise due to PVT behavior and increased nonlinearity. Unlike sourcersink model, the discretized model has the capacity to handle effects of wellbore hydraulics, fluid segregation, and backflow. Pressure drops in horizontal wells, unlike vertical wells, are very important in determining flow from a grid block to a well and flow along the well. Other than near-wellbore phenomenon, grid refinement may play a vital role in improving localized phenomena such as distillation and in-situ upgrading in heavy oil reservoirs compared to fieldwide procedure ŽSharpe et al., 1995..
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
3. Grid orientation Grid orientation effects arise for unfavorable mobility ratio in a displacement process. Grid orientation along with the level of refinement may produce widely varying quantitative simulation results. The use of a diagonal grid may predict steam breakthrough time that is three-fold that predicted using a parallel grid. Abou-Kassem Ž1996. found that for most of the immiscible thermal displacement processes, a nine-point formulation with a two-point upstream mobility alleviates the grid-orientation problem. For miscible problems, a better flood-front tracking is required. Chen et al. Ž1991. and Wolcott et al. Ž1996. have reported successful reduction of grid-orientation problems for both immiscible and miscible processes using higher order differencing techniques. Wolcott et al. Ž1996. pointed out that higher-order techniques ŽHOTs., like the nine-point scheme, significantly reduce grid orientation effects but cannot sharpen the flood-front dispersion profile. At large shock fronts, HOTs create oscillation. Total variation diminishing ŽTVD. schemes use a single point upstream weighting and a higher-order correction factor. Along with the relative permeability and mobility, TVD can be applied for component concentration as well. In a fractional flow situation, the flow rate at i q 1r2 face is a function of concentration gradient ratio Ž r .. Wolcott et al. Ž1996. showed that the equation in such case would be f wn,iq1r2 s f wn,iq1 q
1 2
f iq1r2 Ž f wn,iq1 y f wn,i .
Ž 2.
where f Ž r . is the limiter function. The boundaries for the limiter function can be found from stability analysis. Up to third-order, accurate TVD schemes with a nine-point formulation reduced the grid-orientation error in oil recovery to less than 1% even for coarse grids.
285
processes Ži.e., in-situ combustion, steam stimulation. make the use of three-phase relative permeability more uncertain. In most cases, two-phase relative permeability data are used to obtain three-phase values. It is not possible to find a correlation of water and oil relative permeabilities with water saturation that would be applicable to a wide range of reservoirs. Even with the help of experimental data for the same reservoir, it is difficult to match field production data. Experimental relative permeability values are always suspected in light of the instability theory. The effect of temperature on the water relative permeability end-points is well known ŽFarouq Ali, 1982; Kumar and Do, 1990; Okoye et al., 1990; Tamin and Farouq Ali, 1998.. This must be taken into account for a meaningful history matching. Hysteresis effect on relative permeability in cyclic steam injection Žsteam stimulation. operations should also be considered. Kumar and Do Ž1990. showed that performance prediction of a steam flood was most sensitive to high temperature gas–oil relative permeability near residual oil saturation. They suggested that gas–oil relative permeabilities be measured at steam conditions. Dietrich Ž1981. reported that relative permeability curves generated through numerical simulation to match observed water–oil ratios in several steam stimulation operations worldwide were much lower than those obtained from routine laboratory imbibition water–oil relative permeability tests. End-points measured in the laboratory have been found to match the simulation results. In most simulations, two-phase values are used to obtain three-phase relative permeability. Both Stone’s models and their variations have been widely used for such transformation. Muqeem et al. Ž1995., in their study of three-phase relative permeability measurements, found very little effect of temperature. They, however, reported good matching of the experimental data with Stone’s Method I predictions. All these uncertainties have made relative permeability a frequent history matching parameter.
4. Temperature effects on relative permeability Three-phase relative permeability plays an important role in numerical simulation in general and more so in thermal simulation. The effect of temperature and the complexity of some of the thermal recovery
5. Mathematical models Numerical modeling of thermal-oil-recovery processes reached an advanced stage about a decade back. Abou-Kassem et al. Ž1986. gave a detailed
286
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
description of different aspects of all major models that were developed between 1969 and 1985. The solutions of the reservoir simulation equations of these models are based on the Newton–Raphson method. They gave a comprehensive analysis of mathematical models, treatment of nonlinearities, formulation methods and solution methods of the finite difference equations. The common tendency in the last decade has been to use commercially available simulators that were mostly based on the models reviewed by Abou-Kassem et al. Ž1986.. Therefore, very few attempts have been made to develop new models. One of these efforts has been the volume balance approach. Acs et al. Ž1985. introduced the volume balance principle in modeling a compositional isothermal simulation. For thermal simulation, energy balance provides one more equation per grid block. Mifflin and Watts Ž1991. applied this technique for thermal processes. These investigators used pressure, total mole, and total internal energy as the primary variables. Using total moles Žinstead of mole fractions. and total internal energy as primary variables does away with variable substitutions to handle phase appearance and disappearance. In other words, the flow equations are de-coupled from phase behavior, or simply one can use multiple sets of phase behavior relationships with a single set of flow equations. The terms in volume balance equations have Žstraightforward. physical meanings. For example, the implicit material balance equation for component m written for grid block i in one-dimensional space is expressed as: 1 n nq1 nq1 nq1 Nmnq . ,i y Nm ,i s D t Um ,iy1r2 y Um ,iq1r2 q q m ,i
Ž 3. In fully implicit formulation of a thermal model, a grid block has Nc q 2 unknowns Žtotal moles of Nc components, total internal energy, and pressure. and contributes Nc q 2 equations. Material balance for the individual components and energy balance produce Nc q 1 equations for each grid block. An additional equation for each grid block is obtained by applying the principle of volume balance, which states that fluids in a grid block fill exactly the pore space in a grid block at any time Ži.e., Vp s Vtf .. Brantferger et al. Ž1991. used the volume balance
approach but employed enthalpy rather than total internal energy as a primary variable in the energy balance equation. Farkas and Valko Ž1994. presented a direct IMPES-type volume balance technique for adaptive-implicit steam models. Degrees of implicitness at various grid blocks, computational cost, storage requirement, instability, accuracy of results are a few of the topics of great interest to researchers in their attempts to improve the efficiency of the existing simulation models. Oballa et al. Ž1990. proposed an adaptive-implicit method ŽAIM. with a switching criterion based on numerical stability analysis. They reported a maximum of 62% savings in CPU time over fully implicit formulation. Grabensetter et al. Ž1991. proposed several criteria for the switching between implicit and explicit treatment of grid blocks. Vinsome and Shook Ž1993. used grid block face rather than grid block center to choose the degree of implicitness with respect to changes in a primary variable.
6. Solution methods Abou-Kassem et al. Ž1986. discussed the most commonly used solution methods employed in thermal simulators. After more than a decade, these models are still popular. We will only discuss the solution of the Jacobian here. Successful building and solving the Jacobian is the heart of a simulation task. In building the Jacobian, alignment of primary variables and conservation equations is very important. This helps to make the Jacobian as diagonally dominant as possible. Vinsome and Shook Ž1993. recommended proper alignment of unknowns and equations and suggested that oil pressure, gas saturation, and oil saturation unknowns to be respectively aligned with water component equation, energy equation, and the heavy-hydrocarbon component equation. Cicek and Ertekin Ž1996. noted that the effect of an equationrprimary variable alignment on convergence rate is system-dependent. They suggested that a primary variable should be aligned with the equation that is strongest function of the variable. This criterion may have to be compromised to avoid creating a logarithmically singular Jacobian by
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
choosing an equation that is second- or third-strong function of the variable. In addition, an equationr primary variable alignment must not create a singular Jacobian Žthrough aligning a primary variable with an equation that is not a function of.. Vinsome and Shook Ž1993. also observed that the degree of stability, which can be achieved at large throughputs, is quite sensitive to any errors or approximations in building the Jacobian. They suggested that the Jacobian should be built in as simple fashion as possible.
7. Other developments Within the scope of a paper, discussion of the number of developments is always limited. Several other topics could be discussed more elaborately such as steam distillation in heavy oil reservoirs. It should be recognized as an important oil recovery mechanism in heavy oil reservoirs as it is in light oil reservoirs. Sharpe et al. Ž1995. reported evidence of steam distillation and in-situ upgrading of heavy oil in contrast to the general assumption that the only benefit of steam flooding a heavy oil reservoir is reduction in oil viscosity. They showed that proper study of this mechanism requires an extremely fine grid refinement and accurate equation of state ŽEOS. to represent PVT behavior. Effective permeability for a simulation grid block, whose size is usually 10 to several hundred-fold the size of a geologic model cell, is difficult to determine. Pseudorelative permeabilities are being used to upscale multiphase flow in thermal simulation ŽIto et al., 1993.. Barker and Thibeau Ž1997. presented a critical review on the use of pseudorelative permeability. With the advances of faster computers, vectorization and parallelization are the recent trends in all reservoir simulation including thermal processes. Although vectorization of a solver is possible, parallelization of a solver remains the main stumble block for the parallelization of thermal simulators ŽVinsome and Shook, 1993.. Due to inherent recursiveness, strong pre-conditioners such as ILU factorization or nested factorization are not suitable for parallel computers. Xing and Ma Ž1996. presented a new parallel
287
ILU pre-conditioner based on the technique of sequential staging of tasks ŽSST.. Parallelization has opened the door for achieving tremendous increase in computations. Naccache Ž1997. reported the success of a large, full-field, fine-grid, thermal simulation on a parallel processor. The simulation involves 36,864 cells with 64 steam injectors and 81 producers. The reservoir Žarea wise. was divided into black and white cells and the nested factorization was generated to work within each set of cells. To parallelize the code, the entire simulator was run on each processor independently where it dealt its own set of cells. A message passing protocol was used to communicate between the processors. Using an IBM SP2, the simulation took 13.5 h to run on a single processor but only 23 min on 32 processors, a 35 times speed enhancement. Use of less memory location, efficient use of cache memoryrsystem resources and no use of virtual memory contributed significantly to this improved performance.
8. Conclusions Although advances in computational techniques have given a high level of confidence to thermal simulator users, many more aspects of the processes require further investigation. Thermal effect on rock–fluid interaction is one area where more research is needed to understand the physics of the problem. Characterization of wettability change with temperature is a useful way to lump all the parameters affected by temperature ŽDandina, 1996.. Such parameters include contact angle, interfacial tension, end-point saturations, and oil–water viscosity ratio. Steam stimulation operations in heavy-oil fields with geomechanical changes due to formation parting and in-situ combustion are two processes, which lack the simulation maturity of other thermal methods.
9. Nomenclature fw N Nc p
fractional flow of water total moles of component number of components pressure
288
q r Dt U
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
Vp Vtf fŽr.
molar rate of injection Žproduction. flux ratio time step size molar rate of transport of component across block interface pore volume total fluid volume limiter function
Superscripts n nq1
old time level new time level
Subscripts i y 1r2 i q 1r2 i m p tf
block i interface in the negative direction of x block i interface in the positive direction of x grid block number in x direction component or intermediate time step m pore volume total fluid
References Abou-Kassem, J.H., 1996. Practical considerations in developing numerical simulators for thermal recovery. J. Pet. Sci. Eng. 15 Ž2r4., 281–290. Abou-Kassem, J.H., Farouq Ali, S.M., Ferrer, J., 1986. Appraisal of steamflood models. Rev. Tec. Ing. 9 Ž2., 45–58. Acs, G., Dolesclall, S., Farkas, E., 1985. General purpose compositional model. Soc. Pet. Eng. J. 25 Ž4., 543–552. Babu, D.K., Odeh, A.S., 1989. Productivity of a horizontal well. SPE Res. Eng. 4 Ž3., 417–421. Barker, J.W., Thibeau, S., 1997. A critical review of the use of pseudorelative permeabilities for up scaling. SPE Res. Eng. 12 Ž2., 138–143. Brantferger, R.T., Pope, G.A., Sepehrnoori, K., 1991. Development of a thermodynamically consistent, fully implicit, equation-of-state, compositional steamflood simulator. SPE Pap. 21253, SPE Symp. Res. Simul., Anaheim. Chen, W.H., Durlofsky, L.J., Engquist, B., Osher, S., 1991. Minimization of grid orientation effects through use of higher-order finite difference methods. SPE Pap. 22887, SPE Annu. Tech. Conf. Exhibit., Dallas. Cicek, O., Ertekin, T., 1996. Development and testing of a new 3-D field scale fully implicit multi-phase compositional steam injection simulator. SPE Pap. 35516, SPE Eur. 3-D Res. Modeling Conf., Stavenger, Norway.
Dandina, N.R., 1996. Wettability effects in thermal recovery operations. SPE Pap. 35462, SPErDOE Symp. Improved Oil Rec., Tulsa. Dietrich, J.K., 1981. Relative permeability during cyclic steam stimulation of heavy-oil. J. Pet. Technol. 33 Ž10., 1987–1989. Farkas, E., Valko, P., 1994. A direct IMPES-type volume balance technique for adaptive implicit steam models. SPE Adv. Technol. Ser. 2 Ž2., 88–94, SPE Pap. 26127. Farouq Ali, S.M., 1982. Steam injection theory — a unified approach. SPE Pap. 10746, SPE Calif. Region. Meet., San Francisco. Grabensetter, J., Li, Y.-K., Collins, D.A., Nghiem, L.X., 1991. Stability-based switching criterion for adaptive-implicit compositional reservoir simulation. SPE Pap. 21225, SPE Symp. Res. Simul., Anaheim. Hiebert, A.D., Fung, L.S.-K., Oballa, V., Mourits, F.M., 1993. Comparison of discretization methods for modeling near-well phenomena in thermal processes. J. Can. Pet. Technol. 32 Ž3., 46. Ito, Y., Settari, A., Jha, K., 1993. The effects of shear failure on the cyclic steam process and new pseudo functions for reservoir simulation. J. Can. Pet. Technol. 32 Ž12., 39. Kumar, M., Do, T.N., 1990. Effects of end point saturations and relative permeability models on predicted steamflood performance. SPE Pap. 20202, SPErDOE Symp. Enhanced Oil Rec., Tulsa. Mifflin, R.T., Watts, J.W., 1991. A fully coupled, fully implicit reservoir simulator for thermal and other complex reservoir processes. SPE Pap. 21252, SPE Symp. Res. Simul., Anaheim. Muqeem, M.A., Bentsen, R.G., Maini, B.B., 1995. Effect of temperature on three-phase water–oil–gas relative permeabilities of unconsolidated sand. J. Can. Pet. Technol. 34 Ž3., 34–41. Naccache, P.F., 1997. A fully implicit thermal reservoir simulator. SPE Pap. 37985, SPE Res. Simul. Symp., Dallas. Oballa, V., Coombe, D.A., Buchanan, W.L., 1990. Adaptive-implicit method in thermal simulation. SPE Res. Eng. 5 Ž4., 549–554. Oballa, V., Coombe, D.A., Buchanan, L., 1997. Aspects of discretized wellbore modeling coupled to compositionalrthermal simulation. J. Can. Pet. Technol. 36 Ž4., 45–51. Okoye, C.U., Hayatdavoudi, A., Oungpasuk, P., Wang, P.-H., 1990. The effect of temperature and interfacial tension on oil–water relative permeabilities in consolidated and unconsolidated porous media. SPE Pap. 21067, SPE Latin Am. Petrol. Eng. Conf., Rio de Jenerio. Peaceman, D.W., 1983. Interpretation of well-block pressures in numerical reservoir simulation with nonsquare gridblocks and anisotropic permeability. Soc. Pet. Eng. J. 23 Ž3., 531–543. Rodrigues, J.R.P., Dickstein, F., 1996. A multigrid procedure for local grid refinement in oil reservoir simulation. SPE Adv. Technol. Ser. 4 Ž1., 157–164, SPE Pap. 27049. Sharpe, H.N., Richardson, W.C., Lolley, C.S., 1995. Representation of steam distillation and in-situ upgrading processes in heavy oil simulation. SPE Pap. 30301, SPE Int. Heavy Oil Symp., Calgary. Tamim, M., Farouq Ali, S.M., 1998. A new analytical cyclic
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289 steam stimulation model including formation fracturing. J. Can. Pet. Technol. 37 Ž3., 31–40. Vinsome, P.K.W., Shook, G.M., 1993. Multi-purpose simulation. J. Pet. Sci. Eng. 9 Ž1., 29–38. Wolcott, D.S., Kazemi, H., Dean, R.H., 1996. A practical method for minimizing the grid orientation effect in reservoir simula-
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tion. SPE Pap. 36723, SPE Annu. Tech. Conf. Exhibit., Denver. Xing, J., Ma, Y.L., 1996. A new solution to ILU preconditioning parallelization and its application to reservoir simulations. SPE Pap. 35998, SPE Petrol. Comp. Conf., Dallas.
Recent developments in numerical simulation techniques of thermal recovery processes M. Tamim a , J.H. Abou-Kassem b,) , S.M. Farouq Ali c b
a Bangladesh UniÕersity of Engineering and Technology, Bangladesh Chemical and Petroleum Engineering Department, UAE UniÕersity, Al-Ain 17555, United Arab Emirates c UniÕersity of Alberta, Alberta, Canada
Received 12 December 1998; accepted 15 December 1999
Abstract Numerical simulation of thermal processes Žsteam flooding, steam stimulation, SAGD, in-situ combustion, electrical heating, etc.. is an integral part of a thermal project design. The general tendency in the last 10 years has been to use commercial simulators. During the last decade, only a few new models have been reported in the literature. More work has been done to modify and refine solutions to existing problems to improve the efficiency of simulators. The paper discusses some of the recent developments in simulation techniques of thermal processes such as grid refinement, grid orientation, effect of temperature on relative permeability, mathematical models, and solution methods. The various aspects of simulation discussed here promote better understanding of the problems encountered in the simulation of thermal processes and will be of value to both simulator users and developers. q 2000 Elsevier Science B.V. All rights reserved. Keywords: reservoir simulation; modeling; thermal recovery; developments
1. Introduction Reservoir simulation has gained confidence as evidenced by the frequent use of simulation results in the reservoir engineering decision-making process. Thermal simulators started evolving in the early 1970s. Thermal computational modeling is difficult due to many complex mechanisms including high degree of nonlinearity. Better physical understanding of the mechanisms involved in thermal operations and the advancement of implicit computational tech-
)
Corresponding author.
nique helped thermal simulators to gain maturity. These improvements and successful applications have given the petroleum and software engineers the confidence to develop commercial simulators. This resulted in the transition from command-based formatted data input to colorful graphical user interface. Therefore, the petroleum industry and researchers have moved more towards using commercially available simulators in the last decade. Recent studies are concerned with improving the problem areas of the existing simulation models. Some of these topics and one new modeling approach in thermal numerical simulation technique are discussed in this paper.
0920-4105r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 4 1 0 5 Ž 0 0 . 0 0 0 4 2 - 5
284
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
2. Grid refinement In a thermal process, simulation results are largely influenced by improper modeling of near-wellbore phenomenon and handling of localities where variables change rapidly. Therefore, it is possible to predict inaccurate steam injection pressure or steam override behavior at the wellbore. Coning behavior in thermal operations could also be erroneously predicted due to improper modeling. Local grid refinement is a natural process to handle these situations. In grid refinement procedures, it is common to find irregular connections that evolve from the coupling of coarse and fine grid blocks. This produces matrices with complex structure. Rodrigues and Dickstein Ž1996. addressed this problem using a composite mesh based on a finite volume technique. The grid blocks were distributed among four groups. The first group comprises those blocks in the basic coarse Žparent. mesh that have no connections with refined blocks. The second group comprises those blocks in the refined mesh that have no interface with the basic coarse mesh. These two groups have the usual seven-point operator with additional balance equation for each of the intermediate time steps. The refined blocks at the interface were put in the third group. A refined block ijk and the parent coarse block IJK would have the interface as i q 1r2, j,k. The accumulation, source and flux terms for the other faces remained the same. Pressure at t nq m r M at point x Iq1, jk is required to calculate the pressure gradient at the interface. This point is not in the mesh ŽFig. 1.. Interpolation in time and space would be required. The simplest of the interpolation schemes was used; the interpolation by a constant nq m r M nq 1 pIq1, jk s pIq1, JK
Ž 1.
Boundary coarse blocks were put as the fourth group. Balancing the sum of the fluxes of the refined blocks with their parent coarse block for all intermediate time steps. Their multigrid procedure to solve the set of nonlinear equations allowed for local time stepping. Apart from the abovementioned discretization method ŽCartesian hybrid grid., other methods of discretization such as control volume finite element ŽCVFE. grids and hybrid CVFE grids have also been used to address near-wellbore phenomenon. Rapidly
Fig. 1. Coarse and fine grid interface Žpressure interpolation required at x ŽRodrigues and Dickstein, 1996..
changing variables like temperature, pressure, or saturation require smaller grid blocks mostly around the wellbore. Hiebert et al. Ž1993. found that Cartesian and CVFE hybrid grids have the capacity to accurately model near-well phenomenon. CVFE grid has more flexibility and may provide easier coupling with the cylindrical grid that is used for a more accurate flow modeling around the wellbore. In recent years, improved drilling techniques coupled with lower cost and efficient use have made horizontal drilling more popular in many thermal operations. Integration of the traditional sinkrsource modeling of these wells with a field-wide thermal simulator neglects compositional, energy, and frictional pressure variation along the wellbore. In a discretized wellbore model, each section of the wellbore is treated as a grid block that handles all property variations in a unified manner with the reservoir. Peaceman Ž1983. and Babu and Odeh Ž1989. reported equations for well geometric factors for blocks penetrated by horizontal wells. Oballa et al. Ž1997. explored the advantages and limitations of discretized wellbore models. They recommended that caution be exercised when using the more complex-discretized-wellbore model. Numerical difficulties may arise due to PVT behavior and increased nonlinearity. Unlike sourcersink model, the discretized model has the capacity to handle effects of wellbore hydraulics, fluid segregation, and backflow. Pressure drops in horizontal wells, unlike vertical wells, are very important in determining flow from a grid block to a well and flow along the well. Other than near-wellbore phenomenon, grid refinement may play a vital role in improving localized phenomena such as distillation and in-situ upgrading in heavy oil reservoirs compared to fieldwide procedure ŽSharpe et al., 1995..
M. Tamim et al.r Journal of Petroleum Science and Engineering 26 (2000) 283–289
3. Grid orientation Grid orientation effects arise for unfavorable mobility ratio in a displacement process. Grid orientation along with the level of refinement may produce widely varying quantitative simulation results. The use of a diagonal grid may predict steam breakthrough time that is three-fold that predicted using a parallel grid. Abou-Kassem Ž1996. found that for most of the immiscible thermal displacement processes, a nine-point formulation with a two-point upstream mobility alleviates the grid-orientation problem. For miscible problems, a better flood-front tracking is required. Chen et al. Ž1991. and Wolcott et al. Ž1996. have reported successful reduction of grid-orientation problems for both immiscible and miscible processes using higher order differencing techniques. Wolcott et al. Ž1996. pointed out that higher-order techniques ŽHOTs., like the nine-point scheme, significantly reduce grid orientation effects but cannot sharpen the flood-front dispersion profile. At large shock fronts, HOTs create oscillation. Total variation diminishing ŽTVD. schemes use a single point upstream weighting and a higher-order correction factor. Along with the relative permeability and mobility, TVD can be applied for component concentration as well. In a fractional flow situation, the flow rate at i q 1r2 face is a function of concentration gradient ratio Ž r .. Wolcott et al. Ž1996. showed that the equation in such case would be f wn,iq1r2 s f wn,iq1 q
1 2
f iq1r2 Ž f wn,iq1 y f wn,i .
Ž 2.
where f Ž r . is the limiter function. The boundaries for the limiter function can be found from stability analysis. Up to third-order, accurate TVD schemes with a nine-point formulation reduced the grid-orientation error in oil recovery to less than 1% even for coarse grids.
285
processes Ži.e., in-situ combustion, steam stimulation. make the use of three-phase relative permeability more uncertain. In most cases, two-phase relative permeability data are used to obtain three-phase values. It is not possible to find a correlation of water and oil relative permeabilities with water saturation that would be applicable to a wide range of reservoirs. Even with the help of experimental data for the same reservoir, it is difficult to match field production data. Experimental relative permeability values are always suspected in light of the instability theory. The effect of temperature on the water relative permeability end-points is well known ŽFarouq Ali, 1982; Kumar and Do, 1990; Okoye et al., 1990; Tamin and Farouq Ali, 1998.. This must be taken into account for a meaningful history matching. Hysteresis effect on relative permeability in cyclic steam injection Žsteam stimulation. operations should also be considered. Kumar and Do Ž1990. showed that performance prediction of a steam flood was most sensitive to high temperature gas–oil relative permeability near residual oil saturation. They suggested that gas–oil relative permeabilities be measured at steam conditions. Dietrich Ž1981. reported that relative permeability curves generated through numerical simulation to match observed water–oil ratios in several steam stimulation operations worldwide were much lower than those obtained from routine laboratory imbibition water–oil relative permeability tests. End-points measured in the laboratory have been found to match the simulation results. In most simulations, two-phase values are used to obtain three-phase relative permeability. Both Stone’s models and their variations have been widely used for such transformation. Muqeem et al. Ž1995., in their study of three-phase relative permeability measurements, found very little effect of temperature. They, however, reported good matching of the experimental data with Stone’s Method I predictions. All these uncertainties have made relative permeability a frequent history matching parameter.
4. Temperature effects on relative permeability Three-phase relative permeability plays an important role in numerical simulation in general and more so in thermal simulation. The effect of temperature and the complexity of some of the thermal recovery
5. Mathematical models Numerical modeling of thermal-oil-recovery processes reached an advanced stage about a decade back. Abou-Kassem et al. Ž1986. gave a detailed
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description of different aspects of all major models that were developed between 1969 and 1985. The solutions of the reservoir simulation equations of these models are based on the Newton–Raphson method. They gave a comprehensive analysis of mathematical models, treatment of nonlinearities, formulation methods and solution methods of the finite difference equations. The common tendency in the last decade has been to use commercially available simulators that were mostly based on the models reviewed by Abou-Kassem et al. Ž1986.. Therefore, very few attempts have been made to develop new models. One of these efforts has been the volume balance approach. Acs et al. Ž1985. introduced the volume balance principle in modeling a compositional isothermal simulation. For thermal simulation, energy balance provides one more equation per grid block. Mifflin and Watts Ž1991. applied this technique for thermal processes. These investigators used pressure, total mole, and total internal energy as the primary variables. Using total moles Žinstead of mole fractions. and total internal energy as primary variables does away with variable substitutions to handle phase appearance and disappearance. In other words, the flow equations are de-coupled from phase behavior, or simply one can use multiple sets of phase behavior relationships with a single set of flow equations. The terms in volume balance equations have Žstraightforward. physical meanings. For example, the implicit material balance equation for component m written for grid block i in one-dimensional space is expressed as: 1 n nq1 nq1 nq1 Nmnq . ,i y Nm ,i s D t Um ,iy1r2 y Um ,iq1r2 q q m ,i
Ž 3. In fully implicit formulation of a thermal model, a grid block has Nc q 2 unknowns Žtotal moles of Nc components, total internal energy, and pressure. and contributes Nc q 2 equations. Material balance for the individual components and energy balance produce Nc q 1 equations for each grid block. An additional equation for each grid block is obtained by applying the principle of volume balance, which states that fluids in a grid block fill exactly the pore space in a grid block at any time Ži.e., Vp s Vtf .. Brantferger et al. Ž1991. used the volume balance
approach but employed enthalpy rather than total internal energy as a primary variable in the energy balance equation. Farkas and Valko Ž1994. presented a direct IMPES-type volume balance technique for adaptive-implicit steam models. Degrees of implicitness at various grid blocks, computational cost, storage requirement, instability, accuracy of results are a few of the topics of great interest to researchers in their attempts to improve the efficiency of the existing simulation models. Oballa et al. Ž1990. proposed an adaptive-implicit method ŽAIM. with a switching criterion based on numerical stability analysis. They reported a maximum of 62% savings in CPU time over fully implicit formulation. Grabensetter et al. Ž1991. proposed several criteria for the switching between implicit and explicit treatment of grid blocks. Vinsome and Shook Ž1993. used grid block face rather than grid block center to choose the degree of implicitness with respect to changes in a primary variable.
6. Solution methods Abou-Kassem et al. Ž1986. discussed the most commonly used solution methods employed in thermal simulators. After more than a decade, these models are still popular. We will only discuss the solution of the Jacobian here. Successful building and solving the Jacobian is the heart of a simulation task. In building the Jacobian, alignment of primary variables and conservation equations is very important. This helps to make the Jacobian as diagonally dominant as possible. Vinsome and Shook Ž1993. recommended proper alignment of unknowns and equations and suggested that oil pressure, gas saturation, and oil saturation unknowns to be respectively aligned with water component equation, energy equation, and the heavy-hydrocarbon component equation. Cicek and Ertekin Ž1996. noted that the effect of an equationrprimary variable alignment on convergence rate is system-dependent. They suggested that a primary variable should be aligned with the equation that is strongest function of the variable. This criterion may have to be compromised to avoid creating a logarithmically singular Jacobian by
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choosing an equation that is second- or third-strong function of the variable. In addition, an equationr primary variable alignment must not create a singular Jacobian Žthrough aligning a primary variable with an equation that is not a function of.. Vinsome and Shook Ž1993. also observed that the degree of stability, which can be achieved at large throughputs, is quite sensitive to any errors or approximations in building the Jacobian. They suggested that the Jacobian should be built in as simple fashion as possible.
7. Other developments Within the scope of a paper, discussion of the number of developments is always limited. Several other topics could be discussed more elaborately such as steam distillation in heavy oil reservoirs. It should be recognized as an important oil recovery mechanism in heavy oil reservoirs as it is in light oil reservoirs. Sharpe et al. Ž1995. reported evidence of steam distillation and in-situ upgrading of heavy oil in contrast to the general assumption that the only benefit of steam flooding a heavy oil reservoir is reduction in oil viscosity. They showed that proper study of this mechanism requires an extremely fine grid refinement and accurate equation of state ŽEOS. to represent PVT behavior. Effective permeability for a simulation grid block, whose size is usually 10 to several hundred-fold the size of a geologic model cell, is difficult to determine. Pseudorelative permeabilities are being used to upscale multiphase flow in thermal simulation ŽIto et al., 1993.. Barker and Thibeau Ž1997. presented a critical review on the use of pseudorelative permeability. With the advances of faster computers, vectorization and parallelization are the recent trends in all reservoir simulation including thermal processes. Although vectorization of a solver is possible, parallelization of a solver remains the main stumble block for the parallelization of thermal simulators ŽVinsome and Shook, 1993.. Due to inherent recursiveness, strong pre-conditioners such as ILU factorization or nested factorization are not suitable for parallel computers. Xing and Ma Ž1996. presented a new parallel
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ILU pre-conditioner based on the technique of sequential staging of tasks ŽSST.. Parallelization has opened the door for achieving tremendous increase in computations. Naccache Ž1997. reported the success of a large, full-field, fine-grid, thermal simulation on a parallel processor. The simulation involves 36,864 cells with 64 steam injectors and 81 producers. The reservoir Žarea wise. was divided into black and white cells and the nested factorization was generated to work within each set of cells. To parallelize the code, the entire simulator was run on each processor independently where it dealt its own set of cells. A message passing protocol was used to communicate between the processors. Using an IBM SP2, the simulation took 13.5 h to run on a single processor but only 23 min on 32 processors, a 35 times speed enhancement. Use of less memory location, efficient use of cache memoryrsystem resources and no use of virtual memory contributed significantly to this improved performance.
8. Conclusions Although advances in computational techniques have given a high level of confidence to thermal simulator users, many more aspects of the processes require further investigation. Thermal effect on rock–fluid interaction is one area where more research is needed to understand the physics of the problem. Characterization of wettability change with temperature is a useful way to lump all the parameters affected by temperature ŽDandina, 1996.. Such parameters include contact angle, interfacial tension, end-point saturations, and oil–water viscosity ratio. Steam stimulation operations in heavy-oil fields with geomechanical changes due to formation parting and in-situ combustion are two processes, which lack the simulation maturity of other thermal methods.
9. Nomenclature fw N Nc p
fractional flow of water total moles of component number of components pressure
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q r Dt U
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Vp Vtf fŽr.
molar rate of injection Žproduction. flux ratio time step size molar rate of transport of component across block interface pore volume total fluid volume limiter function
Superscripts n nq1
old time level new time level
Subscripts i y 1r2 i q 1r2 i m p tf
block i interface in the negative direction of x block i interface in the positive direction of x grid block number in x direction component or intermediate time step m pore volume total fluid
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