油藏数值模拟自适应网格法中若干问题研究
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摘要
在石油生产工业中,经常通过注入水或其他流体,驱替地层中的原油,从而达到产油和提高采收率的目的。为了准确预报产量和尽可能提高采收效率,数值分析工具必不可少。因此长期以来,对多孔介质中多相渗流驱替问题的数值研究也一直受到广泛关注。
对于含有相变的多相渗流的数值模拟,通常存在着一个明显的两相区与三相区的分界面,界面附近的物理量变化非常剧烈。自适应网格法就是一种处理带有急剧变化锋面的数值算法。在物理量变化剧烈的区域采用细网格,而在物理量变化缓慢的区域采用粗网格。
本文使用自适应网格法模拟二维均质蒸汽辅助重力驱油过程,并将计算结果与国际上通用的油藏数值模拟软件CMG-STARS的计算结果进行详细的对比。对比结果表明,自适应网格法与CMG-STARS在计算精度上基本一致,而计算效率则明显优于CMG-STARS软件。自适应网格法在计算效率上的优越性,使得对大样本参数条件进行反复计算的周期大大缩短,从而能快速发现油藏开采中的问题,并给出优化策略。因而,自适应网格法在油藏数值模拟中具有巨大的工程实际意义。本文对自适应网格法中存在的若干难点问题进行了全面而深入的研究。
首先,自适应网格法中的网格系统由细到粗分为若干层,我们需要由最精细网格的渗透率计算出各层粗网格的等效渗透率。对于空间无关联的均匀油藏内的流动,每个精细网格内的渗透率都是相同的或是相差不大的,网格变量的粗化可以通过算术平均或调和平均即可达到较高的精度要求。然而对于带有空间关联性的非均匀油藏,算术平均和调和平均在大部分情况下不再有效,如何快速有效地计算粗网格的等效渗透率,是发展复杂地层蒸汽热采稠油自适应网格法的重要科学问题之一。重整化方法就是计算等效渗透率的一种重要方法。对于带有空间关联性的多孔介质,重整化渗透率的概率分布,渗透率方差的标度特性,以及重整化之后的渗透率不动点的变化规律都是我们研究的重点。
我们用相关长度来衡量多孔介质的空间关联性。与空间无关联多孔介质不同的是,带有空间关联性的多孔介质的渗透率方差的对数与重整化次数之间的比值不再是一条直线。在重整化过程中,当相关长度大于重整化尺度时,渗透率的方差将缓慢减少,并且当相关长度远远大于重整化尺度时,标度指数将趋于0。随着重整化次数的逐渐增加,当重整化尺度远远大于相关长度时,标度指数将趋于-1n4,与空间无关联多孔介质重整化过程的统计特性一致。同时,我们还研究了带有相关长度的多孔介质重整化之后的渗透率不动点。如果不同方向的相关长度相等,则不同方向的渗透率不动点完全相同,并与空间无关联多孔介质的渗透率不动点保持一致,即与相关长度的大小无关。当不同方向的相关长度不相等时,无论是理论结果还是数值模拟结果都说明,某一方向的相关长度的数值较大,则对应的渗透率不动点的数值也较大。
为了能够准确快速的计算出三维空间自适应网格法中多层网格的等效渗透率,我们还在二维空间重整化方法(King,1987; Noetinger,1997)的基础上,给出了三维空间中标量形式以及张量形式的等效渗透率的重整化方法。数值计算的结果也表明了这两种重整化方法的计算结果都接近于作为参考值的有限差分的计算结果,特别是张量等效渗透率的计算结果比起标量的更加准确。
其次,我们尝试利用自适应网格法来模拟分布有多种具有不同物理性质(包括骨架性质和孔隙性质)的储层岩石的复杂油藏问题。不同的储层岩石具有不同的相对渗透率曲线和表面张力曲线,各相流体的相饱和度在不同储层岩石的交界面处将发生空间间断,如果继续用各相饱和度作为网格系统的划分依据的话,将导致精细网格的数量大幅增加,影响计算的效率。在文中,我们从网格的各界面流量守恒性出发,并作如下假定：忽略毛管力；各相流量在粗网格中的变化是相当小的；同一个粗网格内同种岩石的各相饱和度是一致的。在这些假定的前提下,给出了二维空间多种岩石自适应网格法的粗网格细化以及细网格粗化的新准则,并取得了不错的数值模拟结果。
最后,我们对基础的油水两相等温渗流的数值算法进行了初步的探讨。描述饱和度变化的连续方程,在数学上对应非线性双曲型方程,为了避免中心差分算法带来的物理上不真实的振荡,在通常的数值模拟中我们都采用了迎风格式。然而迎风方法只有一阶精度,会导致较明显的数值耗散。同时控制方程的离散,均采用五点差分格式,这在高维空间中的某些情况下将会导致严重的网格取向效应。虽然可以通过采用九点差分格式、PEBI网格等来减少网格取向效应,但是尚没有统一的数值算法可以有效避免各种情况下出现的网格取向效应。为了尽量减少数值耗散,降低网格取向效应,我们对基础的油水两相等温渗流的数值算法进行了初步的探讨,通过利用特征线方法分析Buckley-Leverett方程,提出了活塞式驱替和非活塞式驱替的严格数学划分条件。由于非活塞式驱替的饱和度在全场连续,本文建议对网格界面处相对渗透率的离散值不采用常用的在迎风方向取值的做法,而直接取相邻网格的算术平均值。数值计算结果表明,对非活塞式驱替而言,该算法不仅可以提高计算精度,而且可以减小网格取向效应。
综上,本文的工作是从自适应网格法的计算结果出发,对自适应网格法中存在的若干难点问题进行研究,为后期的复杂油藏的数值模拟打好坚实的理论基础。相应的数值算例也验证了我们的理论。我们期望通过对油藏数值模拟中自适应网格法的基础问题的研究,能够给以后的工作带来实际的帮助。
In the oil production industry, the crude oil was displacement by water or other fluids which injected into the underground, in order to improve the recovery percent. Numerical analysis tools are essential for forecasting the oil production and improving the recovery percent. Oil-water displacement is an important problem often encountered in the oil field development.
The difficulty for the numerical simulations of multi-phase flow in porous media is the existence of very sharp temperature and saturation fronts around the phase change regions. Adaptive mesh refinement technique is a kind of numerical algorithm dealing with jumpy frontal surface, which is capable of using fine grids in the area with steep gradients but coarse grids where the variations of variables are slower, to track the moving fronts inside the calculation domain. The fine grids use short time step, and the coarse grids use long time step.
In this paper, we applied adaptive mesh refinement(AMR) technique into two-dimensional homogeneous steam assisted gravity flooding(SAGD) process, and compared the results with STARS software, which was widely used in the oil department. The numerical results show that, the accuracy of AMR technique is the same as STARS software, but AMR technique takes less calculation time than STARS software. Therefore, it enables us to implement large amount of simulations rapidly with different parameters. It is of help to find out problems and make better decisions for petroleum recovery. In this paper, we researched on some problems of AMR for reservoir numerical simulation.
First of all, in the AMR technique, we need to calculate the equivalent permeability of the multi-grid. In the isotropy porous media, the equivalent permeability can be calculated easily by arithmetic mean or harmonic mean. But in the anisotropy porous media, arithmetic mean and harmonic mean may no longer be valid, and we must find a new numerical algorithm to calculate the equivalent permeability. Renormalization is a simple and fast method for calculating the equivalent permeability.
The statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated. In contrast to the uncorrelated porous media, scaling behavior of the variance of the renormalized permeability exhibits a crossover behavior. When the correlation lengths are larger compared with the domain scale covered by the renormalization procedure, the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero. As the renormalization number increases, the covered domain scale will eventually become larger than the correlation lengths, and the scaling property will transit to be the same as the uncorrelated case. The convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated. Both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction. For the log-normal permeability field, numerical simulations show that the crossover scaling and the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions.
Based on renormalization methods under two dimensions, two upscaling methods, one obtaining scalar equivalent permeability and the other leading to tensorial equivalent permeability, are generalized to three dimensions. The numerical results indicate that both methods are close to the solutions from finite difference method, while the tensorial one is suggested to be more accurate compared with the scalar one.
Secondly, Adaptive mesh refinement technique was also applied to the non-isothermal multiphase flow in the complex reservoir containing different types of rocks. The reservoir has rocks with different relative permeability curves, and the saturation is disconnected at the interface across different types of rocks. In order to apply AMR technique, we make the following assumptions:the capillary force can be ignored; the change of flows for each phase in coarse grid is very small; each phase saturation with the same rock type in coarse grid is the same. Under these assumptions, we have a new coarsening criterion and a new refining method in the AMR process which has solved the complex reservoir containing different types of rocks and achieved very good numerical simulation results.
At last, we investigate the numerical algorithm of two-phase isothermal flow. The continuity equation is corresponding to the hyperbolic equation, and we usually use upwind scheme in the numerical algorithm for the purpose of avoiding unrealistic physical oscillation. But the first-order accuracy of upwind scheme may result in significant numerical dissipation. And five-point scheme may bring serious grid orientation effect in high-dimensional space. In order to reduce the numerical dissipation and decrease the grid orientation effect, we investigate the numerical algorithm of two-phase isothermal flow. Based upon the analysis of Buckley-Leverett equation by the characteristic line method, a mathematical judging condition is proposed to distinguish between the piston-like displacement and the non-piston-like displacement. Due to that the spatial distributions of the phase saturations keep the continuity, for the non-piston-like displacement, it is suggested that the internodal transmissibility be evaluated by the arithmetic average of the transmissibilities on its neighbor grids rather than by the upstream value. Numerical examples show that the proposed algorithm for the non-piston-like displacement can not only improve the numerical accuracy but also decrease the grid orientation effect.
In summary, several basic problems of AMR technique have been discussed in this paper. The numerical results indicate a good proving for the theory. We expect the research to have a practical help to the reservoir simulation in the future.
对于含有相变的多相渗流的数值模拟,通常存在着一个明显的两相区与三相区的分界面,界面附近的物理量变化非常剧烈。自适应网格法就是一种处理带有急剧变化锋面的数值算法。在物理量变化剧烈的区域采用细网格,而在物理量变化缓慢的区域采用粗网格。
本文使用自适应网格法模拟二维均质蒸汽辅助重力驱油过程,并将计算结果与国际上通用的油藏数值模拟软件CMG-STARS的计算结果进行详细的对比。对比结果表明,自适应网格法与CMG-STARS在计算精度上基本一致,而计算效率则明显优于CMG-STARS软件。自适应网格法在计算效率上的优越性,使得对大样本参数条件进行反复计算的周期大大缩短,从而能快速发现油藏开采中的问题,并给出优化策略。因而,自适应网格法在油藏数值模拟中具有巨大的工程实际意义。本文对自适应网格法中存在的若干难点问题进行了全面而深入的研究。
首先,自适应网格法中的网格系统由细到粗分为若干层,我们需要由最精细网格的渗透率计算出各层粗网格的等效渗透率。对于空间无关联的均匀油藏内的流动,每个精细网格内的渗透率都是相同的或是相差不大的,网格变量的粗化可以通过算术平均或调和平均即可达到较高的精度要求。然而对于带有空间关联性的非均匀油藏,算术平均和调和平均在大部分情况下不再有效,如何快速有效地计算粗网格的等效渗透率,是发展复杂地层蒸汽热采稠油自适应网格法的重要科学问题之一。重整化方法就是计算等效渗透率的一种重要方法。对于带有空间关联性的多孔介质,重整化渗透率的概率分布,渗透率方差的标度特性,以及重整化之后的渗透率不动点的变化规律都是我们研究的重点。
我们用相关长度来衡量多孔介质的空间关联性。与空间无关联多孔介质不同的是,带有空间关联性的多孔介质的渗透率方差的对数与重整化次数之间的比值不再是一条直线。在重整化过程中,当相关长度大于重整化尺度时,渗透率的方差将缓慢减少,并且当相关长度远远大于重整化尺度时,标度指数将趋于0。随着重整化次数的逐渐增加,当重整化尺度远远大于相关长度时,标度指数将趋于-1n4,与空间无关联多孔介质重整化过程的统计特性一致。同时,我们还研究了带有相关长度的多孔介质重整化之后的渗透率不动点。如果不同方向的相关长度相等,则不同方向的渗透率不动点完全相同,并与空间无关联多孔介质的渗透率不动点保持一致,即与相关长度的大小无关。当不同方向的相关长度不相等时,无论是理论结果还是数值模拟结果都说明,某一方向的相关长度的数值较大,则对应的渗透率不动点的数值也较大。
为了能够准确快速的计算出三维空间自适应网格法中多层网格的等效渗透率,我们还在二维空间重整化方法(King,1987; Noetinger,1997)的基础上,给出了三维空间中标量形式以及张量形式的等效渗透率的重整化方法。数值计算的结果也表明了这两种重整化方法的计算结果都接近于作为参考值的有限差分的计算结果,特别是张量等效渗透率的计算结果比起标量的更加准确。
其次,我们尝试利用自适应网格法来模拟分布有多种具有不同物理性质(包括骨架性质和孔隙性质)的储层岩石的复杂油藏问题。不同的储层岩石具有不同的相对渗透率曲线和表面张力曲线,各相流体的相饱和度在不同储层岩石的交界面处将发生空间间断,如果继续用各相饱和度作为网格系统的划分依据的话,将导致精细网格的数量大幅增加,影响计算的效率。在文中,我们从网格的各界面流量守恒性出发,并作如下假定：忽略毛管力；各相流量在粗网格中的变化是相当小的；同一个粗网格内同种岩石的各相饱和度是一致的。在这些假定的前提下,给出了二维空间多种岩石自适应网格法的粗网格细化以及细网格粗化的新准则,并取得了不错的数值模拟结果。
最后,我们对基础的油水两相等温渗流的数值算法进行了初步的探讨。描述饱和度变化的连续方程,在数学上对应非线性双曲型方程,为了避免中心差分算法带来的物理上不真实的振荡,在通常的数值模拟中我们都采用了迎风格式。然而迎风方法只有一阶精度,会导致较明显的数值耗散。同时控制方程的离散,均采用五点差分格式,这在高维空间中的某些情况下将会导致严重的网格取向效应。虽然可以通过采用九点差分格式、PEBI网格等来减少网格取向效应,但是尚没有统一的数值算法可以有效避免各种情况下出现的网格取向效应。为了尽量减少数值耗散,降低网格取向效应,我们对基础的油水两相等温渗流的数值算法进行了初步的探讨,通过利用特征线方法分析Buckley-Leverett方程,提出了活塞式驱替和非活塞式驱替的严格数学划分条件。由于非活塞式驱替的饱和度在全场连续,本文建议对网格界面处相对渗透率的离散值不采用常用的在迎风方向取值的做法,而直接取相邻网格的算术平均值。数值计算结果表明,对非活塞式驱替而言,该算法不仅可以提高计算精度,而且可以减小网格取向效应。
综上,本文的工作是从自适应网格法的计算结果出发,对自适应网格法中存在的若干难点问题进行研究,为后期的复杂油藏的数值模拟打好坚实的理论基础。相应的数值算例也验证了我们的理论。我们期望通过对油藏数值模拟中自适应网格法的基础问题的研究,能够给以后的工作带来实际的帮助。
In the oil production industry, the crude oil was displacement by water or other fluids which injected into the underground, in order to improve the recovery percent. Numerical analysis tools are essential for forecasting the oil production and improving the recovery percent. Oil-water displacement is an important problem often encountered in the oil field development.
The difficulty for the numerical simulations of multi-phase flow in porous media is the existence of very sharp temperature and saturation fronts around the phase change regions. Adaptive mesh refinement technique is a kind of numerical algorithm dealing with jumpy frontal surface, which is capable of using fine grids in the area with steep gradients but coarse grids where the variations of variables are slower, to track the moving fronts inside the calculation domain. The fine grids use short time step, and the coarse grids use long time step.
In this paper, we applied adaptive mesh refinement(AMR) technique into two-dimensional homogeneous steam assisted gravity flooding(SAGD) process, and compared the results with STARS software, which was widely used in the oil department. The numerical results show that, the accuracy of AMR technique is the same as STARS software, but AMR technique takes less calculation time than STARS software. Therefore, it enables us to implement large amount of simulations rapidly with different parameters. It is of help to find out problems and make better decisions for petroleum recovery. In this paper, we researched on some problems of AMR for reservoir numerical simulation.
First of all, in the AMR technique, we need to calculate the equivalent permeability of the multi-grid. In the isotropy porous media, the equivalent permeability can be calculated easily by arithmetic mean or harmonic mean. But in the anisotropy porous media, arithmetic mean and harmonic mean may no longer be valid, and we must find a new numerical algorithm to calculate the equivalent permeability. Renormalization is a simple and fast method for calculating the equivalent permeability.
The statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated. In contrast to the uncorrelated porous media, scaling behavior of the variance of the renormalized permeability exhibits a crossover behavior. When the correlation lengths are larger compared with the domain scale covered by the renormalization procedure, the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero. As the renormalization number increases, the covered domain scale will eventually become larger than the correlation lengths, and the scaling property will transit to be the same as the uncorrelated case. The convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated. Both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction. For the log-normal permeability field, numerical simulations show that the crossover scaling and the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions.
Based on renormalization methods under two dimensions, two upscaling methods, one obtaining scalar equivalent permeability and the other leading to tensorial equivalent permeability, are generalized to three dimensions. The numerical results indicate that both methods are close to the solutions from finite difference method, while the tensorial one is suggested to be more accurate compared with the scalar one.
Secondly, Adaptive mesh refinement technique was also applied to the non-isothermal multiphase flow in the complex reservoir containing different types of rocks. The reservoir has rocks with different relative permeability curves, and the saturation is disconnected at the interface across different types of rocks. In order to apply AMR technique, we make the following assumptions:the capillary force can be ignored; the change of flows for each phase in coarse grid is very small; each phase saturation with the same rock type in coarse grid is the same. Under these assumptions, we have a new coarsening criterion and a new refining method in the AMR process which has solved the complex reservoir containing different types of rocks and achieved very good numerical simulation results.
At last, we investigate the numerical algorithm of two-phase isothermal flow. The continuity equation is corresponding to the hyperbolic equation, and we usually use upwind scheme in the numerical algorithm for the purpose of avoiding unrealistic physical oscillation. But the first-order accuracy of upwind scheme may result in significant numerical dissipation. And five-point scheme may bring serious grid orientation effect in high-dimensional space. In order to reduce the numerical dissipation and decrease the grid orientation effect, we investigate the numerical algorithm of two-phase isothermal flow. Based upon the analysis of Buckley-Leverett equation by the characteristic line method, a mathematical judging condition is proposed to distinguish between the piston-like displacement and the non-piston-like displacement. Due to that the spatial distributions of the phase saturations keep the continuity, for the non-piston-like displacement, it is suggested that the internodal transmissibility be evaluated by the arithmetic average of the transmissibilities on its neighbor grids rather than by the upstream value. Numerical examples show that the proposed algorithm for the non-piston-like displacement can not only improve the numerical accuracy but also decrease the grid orientation effect.
In summary, several basic problems of AMR technique have been discussed in this paper. The numerical results indicate a good proving for the theory. We expect the research to have a practical help to the reservoir simulation in the future.
引文
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