双水平集逼近油藏模型特征的数值模拟方法
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摘要
本文使用双水平集函数逼近油藏模型特征,构造出Uzawas算法进行数值模拟.对于两相流渗透率的数值求解问题,可以通过测量油井数据和地震波数据来实现.将构造出来的带限制的最优化问题使用变异的Lagrange方法求解.如果使用双水平集函数逼近渗透率函数,则需要对Lagrange函数进行修正,从而将带限制的最优化问题转化成无限制的最优化问题.由于双水平集函数的优越性,进一步构造出最速梯度下降Uzawas算法和算子分裂格式Uzawas算法进行求解对应的最优化子问题.数值算例表明设计的算法是高效的、稳定的.
In this paper,Uzawas algorithms by steepest gradient descent scheme and operator splitting scheme for dynamic reservoir characterization by a binary level set approach are explored.The permeability of two-phase porous media flow can be recovered by the augmented Lagrangian optimization method with well data and seismic time-lapse data.After adding the constraint functions to the augmented Lagrangian optimization functional,the saddle points can be found by steepest gradient descent scheme and operator splitting scheme,which is based on the essence of binary level set method with few difficulties.In the end,the numerical examples demonstrate that the given algorithms will be efficient,stable when the permeability is recovered.
In this paper,Uzawas algorithms by steepest gradient descent scheme and operator splitting scheme for dynamic reservoir characterization by a binary level set approach are explored.The permeability of two-phase porous media flow can be recovered by the augmented Lagrangian optimization method with well data and seismic time-lapse data.After adding the constraint functions to the augmented Lagrangian optimization functional,the saddle points can be found by steepest gradient descent scheme and operator splitting scheme,which is based on the essence of binary level set method with few difficulties.In the end,the numerical examples demonstrate that the given algorithms will be efficient,stable when the permeability is recovered.
引文
1Aanonsen S I,Eydinov D.A multiscale method for distributed parameter estimation with application to reservoir history matching.Comput Geosci,2006,1:97-117
2Wang S G,Zhao G Z,Xu L B,et al.Optimization for automatic history matching.Int J Numer Anal Model,2005,2:131-137
3Zhu J P,Chen Y M.Multilevel grid method for history matching multidimensional multiphase reservoir models.Appl Numer Math,1992,10:159-174
4Tarantola A.Inverse Problem Theory and Methods for Model Parameter Etimation.Philadelphia:Society for Indus-trial and Applied Mathematics(SIAM),2005
5Grimstad A-A,Mannseth T,N vdal G,et al.Adaptive multiscale permeability estimation.Comput Geosci,2003,7:1-25
6Berre I,Lien M,Mannseth T.A level-set corrector to an adaptive multiscale permeability prediction.Comput Geosci,2007,1:27-42
7Lien M,Berre I,Mannseth T.Combined adaptive multiscale and level-set parameter estimation.Multiscale Model Simul,2005,4:1349-1372
8Jahn H O.A rapid method for obtaining a two-dimensional reservoir describption from well pressure response data.Soc Petrol Eng J,1966,237:315-327
9Jacquard P,Jain C.Permeability distribution from field pressure data.Soc Petrol Eng J,1965,5:281-294
10Chavent G,Dupuy M,Lemmonier C.Histroy matching by using of optimal control theory.Soc Petrol Eng J,1975,15:74-86
11Gavalas G R,Seinfeld J H.Reservoir history matching by Bayesian estimation.Soc Petrol Eng J,1976,12:337-350
12Kulkarni K N,Datta-Gupta A.Estimating relative permeability from production data:a streamline approach.Soc Petrol Eng J,2000,5:402-411
13Wang Y,Kovscek A R.Streamline approach for history matching production data.Soc Petrol Eng J,2000,5:353-362
14Agarwal B,Blunt M.A streamline-based method for assisted history matching applied to an Arabian Gulf field.Soc Petrol Eng J,2003,9:437-449
15He Z,Datta-Gupta A,Yoon S.Streamline-based production data integration with gravity and changing fields condi-tions.Soc Petrol Eng J,2002,5:423-436
16Caers J.Annual Technical Conference and Exhibition.SPE,2002,77429
17Harding T J,Radciffe N J,King P R.Optimization of productiion strategies using stochastic search methods.SPE European3-D Reservoir Modelling Conference,1996,35518
18Gonz′alez-Rodr′guez P,Kindelan M,Moscoso M,et al.History matching problem in reservoir engineering using the propagation-backpropagation method.Inverse Problems,2005,2:565-590
19He Z,Lee T Y,Seinfeld J H.Estimation of two-phase petroleum reservoir properties by regularization.J Comput Phys,1987,2:397-419
20Wasserman M L,Emanuel A S,Seinfeld J H.Pratical applications of optiaml-cpntrol theory to history-matching multiphases simulator models.Soc Petrol Eng J,1975,15:347-355
21Wu Z,Reynolds A C,Oliver D S.Conditioning geostatical models to two-phase production data.Soc Petrol Eng J,1999,4:142-155
22Lie J,Lysaker M,Tai X C.Piecewise Constant Level Set Methods and Image Segmentation.Scale Space and PDE Methods in Computer Vision:5th International Conference.2005,573-584
23Lie J,Lysaker M,Tai X C.A variant of the level set method and applications to image segmentatio.Math Comp,2006,75:1155-1174
24Tai X C,Yao C H.Image segmentation by piecewise constant Mumford-Shah model without estimating the constants.J Comput Math,2006,24:435-443
25Lie J,Lysaker M,Tai X C.A binary level set model and some applications for Mumford-Shah image segmentation.IEEE Trans Image Proc,2006,15:423-436
26Mumford D,Shah J.Optimal approximations by piecewise smooth functions and associated variational problems.Comm Pure Appl Math,1989,42:577-685
27Chan T,Vese L A.Active contours without edges.IEEE Trans Image Proc,2001,10:266-277
28Chan T F,Tai X C.Identification of discontinuous coefficients in elliptic problems using total variation regularization.SIAM J Sci Comput,2003,25:881-904
29Tai X C,Chan T F.A survey on multiple level set methods with applications for identifying piecewise constant functions.Int J Numer Anal Model,2004,1:25-47
30Sethian J A.Level set methods and fast marching methods.Cambridge Monographs on Applied and Computational Mathematics,1999
31Tsai R,Osher S.Level set methods and their applications in image science.Comm Math Sci,2003,1:623-656
32IAanonsen S,Cominelli A,Gosselin O,et al.Integrationn of4D data in the history match loop by investigating scale dependent correlations in the acustic impedence cube.In:Proceeding of the8th European Conference on Mathematics of Oil Recovery,2002
33Li H W,Tai X C.Piecewise constant level set method for multiphase motion.Int J Numer Anal Model,2007,4:291-305
34Nielsen L K,Tai X C,Aanonsen S I,et al.A binary level set model for elliptic inverse problems with discontinuouscoefficients.Int J Numer Anal Model,2007,4:74-99
35Lu T,Neittaanma¨ki P,Tai X C.A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations.RAIRO Mod′el Math Anal Num′er,1992,26:1992
36Li R,Reynolds A C,Oliver D S.Piecewise constant level set method for multiphase motion.Int J Numer Anal Model,2007,4:291-305
2Wang S G,Zhao G Z,Xu L B,et al.Optimization for automatic history matching.Int J Numer Anal Model,2005,2:131-137
3Zhu J P,Chen Y M.Multilevel grid method for history matching multidimensional multiphase reservoir models.Appl Numer Math,1992,10:159-174
4Tarantola A.Inverse Problem Theory and Methods for Model Parameter Etimation.Philadelphia:Society for Indus-trial and Applied Mathematics(SIAM),2005
5Grimstad A-A,Mannseth T,N vdal G,et al.Adaptive multiscale permeability estimation.Comput Geosci,2003,7:1-25
6Berre I,Lien M,Mannseth T.A level-set corrector to an adaptive multiscale permeability prediction.Comput Geosci,2007,1:27-42
7Lien M,Berre I,Mannseth T.Combined adaptive multiscale and level-set parameter estimation.Multiscale Model Simul,2005,4:1349-1372
8Jahn H O.A rapid method for obtaining a two-dimensional reservoir describption from well pressure response data.Soc Petrol Eng J,1966,237:315-327
9Jacquard P,Jain C.Permeability distribution from field pressure data.Soc Petrol Eng J,1965,5:281-294
10Chavent G,Dupuy M,Lemmonier C.Histroy matching by using of optimal control theory.Soc Petrol Eng J,1975,15:74-86
11Gavalas G R,Seinfeld J H.Reservoir history matching by Bayesian estimation.Soc Petrol Eng J,1976,12:337-350
12Kulkarni K N,Datta-Gupta A.Estimating relative permeability from production data:a streamline approach.Soc Petrol Eng J,2000,5:402-411
13Wang Y,Kovscek A R.Streamline approach for history matching production data.Soc Petrol Eng J,2000,5:353-362
14Agarwal B,Blunt M.A streamline-based method for assisted history matching applied to an Arabian Gulf field.Soc Petrol Eng J,2003,9:437-449
15He Z,Datta-Gupta A,Yoon S.Streamline-based production data integration with gravity and changing fields condi-tions.Soc Petrol Eng J,2002,5:423-436
16Caers J.Annual Technical Conference and Exhibition.SPE,2002,77429
17Harding T J,Radciffe N J,King P R.Optimization of productiion strategies using stochastic search methods.SPE European3-D Reservoir Modelling Conference,1996,35518
18Gonz′alez-Rodr′guez P,Kindelan M,Moscoso M,et al.History matching problem in reservoir engineering using the propagation-backpropagation method.Inverse Problems,2005,2:565-590
19He Z,Lee T Y,Seinfeld J H.Estimation of two-phase petroleum reservoir properties by regularization.J Comput Phys,1987,2:397-419
20Wasserman M L,Emanuel A S,Seinfeld J H.Pratical applications of optiaml-cpntrol theory to history-matching multiphases simulator models.Soc Petrol Eng J,1975,15:347-355
21Wu Z,Reynolds A C,Oliver D S.Conditioning geostatical models to two-phase production data.Soc Petrol Eng J,1999,4:142-155
22Lie J,Lysaker M,Tai X C.Piecewise Constant Level Set Methods and Image Segmentation.Scale Space and PDE Methods in Computer Vision:5th International Conference.2005,573-584
23Lie J,Lysaker M,Tai X C.A variant of the level set method and applications to image segmentatio.Math Comp,2006,75:1155-1174
24Tai X C,Yao C H.Image segmentation by piecewise constant Mumford-Shah model without estimating the constants.J Comput Math,2006,24:435-443
25Lie J,Lysaker M,Tai X C.A binary level set model and some applications for Mumford-Shah image segmentation.IEEE Trans Image Proc,2006,15:423-436
26Mumford D,Shah J.Optimal approximations by piecewise smooth functions and associated variational problems.Comm Pure Appl Math,1989,42:577-685
27Chan T,Vese L A.Active contours without edges.IEEE Trans Image Proc,2001,10:266-277
28Chan T F,Tai X C.Identification of discontinuous coefficients in elliptic problems using total variation regularization.SIAM J Sci Comput,2003,25:881-904
29Tai X C,Chan T F.A survey on multiple level set methods with applications for identifying piecewise constant functions.Int J Numer Anal Model,2004,1:25-47
30Sethian J A.Level set methods and fast marching methods.Cambridge Monographs on Applied and Computational Mathematics,1999
31Tsai R,Osher S.Level set methods and their applications in image science.Comm Math Sci,2003,1:623-656
32IAanonsen S,Cominelli A,Gosselin O,et al.Integrationn of4D data in the history match loop by investigating scale dependent correlations in the acustic impedence cube.In:Proceeding of the8th European Conference on Mathematics of Oil Recovery,2002
33Li H W,Tai X C.Piecewise constant level set method for multiphase motion.Int J Numer Anal Model,2007,4:291-305
34Nielsen L K,Tai X C,Aanonsen S I,et al.A binary level set model for elliptic inverse problems with discontinuouscoefficients.Int J Numer Anal Model,2007,4:74-99
35Lu T,Neittaanma¨ki P,Tai X C.A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations.RAIRO Mod′el Math Anal Num′er,1992,26:1992
36Li R,Reynolds A C,Oliver D S.Piecewise constant level set method for multiphase motion.Int J Numer Anal Model,2007,4:291-305
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