复杂地层油藏蒸汽注采数值模拟自适应网格法的研究
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摘要
蒸汽注采是稠油油藏开发的主要手段。对于复杂地层油藏蒸汽注采过程的数值模拟,其中一个困难在于计算流场中存在一个温度、饱和度剧烈变化的相变锋面。在锋面附近位置,物理量变化非常剧烈,计算网格的尺寸必须很小以满足计算精度的要求,如果对计算区域采用均匀网格计算,则计算量将难以忍受；另外,由于地层的复杂性,往往采用非结构化网格处理,这势必会增加数值离散上的复杂度,而若采用结构化网格,其尺度又会影响复杂区域的计算精度。因此,有必要建立高效而又精确的数值模拟手段,自适应网格法就是本文的主要研究内容。
本文工作的目标是将自适应网格法应用于复杂地层稠油油藏蒸汽注采过程的数值模拟,并研究引入自适应网格法后遇到的相关数学物理问题。
对均质油藏的自适应网格展开研究。利用结构化程序设计方案编写蒸汽注采自适应网格法程序,结构化的程序能够很方便地将后续研究工作引入,从而大幅度缩减数值程序的编写时间。二维与三维的均质油藏蒸汽辅助重力泄油(SAGD)算例显示,自适应网格法能大幅度提高模拟的计算速度,并且保持很高的计算精度。自适应网格结构也显示计算区域中的精细网格数量大幅度减少。
在将自适应网格技术应用到非均质油藏的模拟中时,对于自适应网格系统中的各层次粗网格,采用重正化方法计算等效渗透率。为削减网格取向效应的影响,对二维蒸汽驱的模拟采用九点离散格式,并建立自适应网格系统下非均质油藏的控制方程的离散方法。二维非均质油藏SAGD算例和蒸汽驱算例显示,在此类油藏的模拟中,自适应网格法也能保持很高的计算精度,并且能大幅度提高计算速度。
对于复杂断层油藏的模拟,本文提出将断层处理成渗透率极低的介质,从而将断层油藏类比于非均质油藏,对整个断层油藏划分均匀的结构化网格。对于网格系统中被断层穿过的网格,文中对其等效渗透率的处理提出按流体流通通道进行计算的方法,而不是直接将网格的等效渗透率置为极低值。对于同时被断层穿过的相邻网格,本文提出,在计算它们之间的流量时,必须考虑断层连续性的影响。对断层油藏划分结构化网格后,将自适应网格技术应用其中。自适应网格系统中网格粗化仅依据所选物理量的空间变化,不专门针对断层附近区域划分精细网格。断层区域将依据其附近物理量的变化适时地采用较粗网格计算,从而进一步提高自适应网格法程序的计算速度。对于断层区域附近各层次粗网格,采用重正化方法计算其等效渗透率。二维SAGD算例及蒸汽驱算例表明,将断层处理成渗透介质没有破坏断层本身的物理性质,而自适应网格法的引入大幅度提升了断层油藏模拟的计算速度,同时与全精细网格法相比,自适应网格法也保持了很高的计算精度。
在对复杂边界油藏进行数值研究时,结构化网格往往难以划分。本文提出以一块较大的矩形区域覆盖整个计算区域,针对该矩形区域划分结构化网格的办法。网格结构中,位于边界外的网格并不参与计算,整个计算复杂度没有增加。为充分考虑边界的影响,本文采用界面渗透率代替网格渗透率进行数值计算的方案,对被边界穿过的界面提出渗透率的预处理方法。单相流的验证算例显示,该方案与预处理方法对于复杂边界的模拟能够取得很高的计算精度,即使在网格尺度较大的情况下,计算所得的解依然很接近收敛解。将自适应网格技术应用到复杂边界油藏的模拟中时,网格系统的粗化判据仅为所选物理量的空间变化,并不考虑网格是否位于边界。在计算过程中,计算边界附近会适时的采用粗网格,从而大幅度提高了计算速度。对于边界处的各层次粗网格,采用和精细网格同样的预处理方法计算其界面等效渗透率。二维水平井蒸汽驱算例以及SAGD算例显示,自适应网格法能保持很高计算速度,边界处适时地采用粗网格计算并不影响自适应网格法的计算精度。将文中方案与预处理方法推广到三维问题的计算,三维复杂地层地下水源热泵问题的模拟也显示出AMR计算效率与计算精度上的优势。
综上,本文的工作是对复杂地层稠油油藏蒸汽注采数值模拟自适应网格算法及实施自适应网格算法时遇到的数学物理问题展开研究。文中的数值算例显示自适应网格算法在计算效率上有着巨大的优势,同时在计算精度上与全精细网格法非常吻合。希望自适应网格法能够对生产现场的复杂地层油藏蒸汽注采数值模拟带来帮助。
Steam injection is commonly used to recover heavy oils from petroleum reservoirs. Numerical simulation is a very difficult task, since high nonlinearity in the problem formulation may lead to very sharp thermal and saturation fronts. Because of these rapid variations of the physical quantities across the fronts, numerical grids must be fine enough to achieve reasonable accuracy. As a result, huge CPU time and memory are needed if applying a uniform fine grid to the whole calculation area. In addition, due to the complexity of the reservoir, unstructured meshes are widely used, which is bound to increase the complexity of the numerical discrete. If we use structured grids, the scale should be fine enough to satisfy the simulation accuracy of the complex region. Therefore, improving the calculation efficiency for such problems has great theoretical and practical interest, and the adaptive mesh refinement(AMR) technique is a class of strategies that address this problem.
In this paper, we present an application of the AMR technique to the numerical simulation of steam thermal recovery processes in complex petroleum reservoirs. And several problems relevant to the AMR numerical simulations are carefully studied.
For homogeneous reservoirs, we suggest using structured program design to write the AMR programs. We can easily check the flow-up studies and the preparation time of the numerical procedure can be substantially reduced. The numerical examples for2D and3D SAGD process shows that, the AMR technique results are fast compared with the solutions under referenced uniformly fine grids, and can give good accuracy. AMR grid structure shows that the number of fine grids of the reservoir is significantly reduced.
For heterogeneous reservoirs, effective permeability of the coarse grids must be calculated while we apply the AMR techniques, and the renormalization method is adopted. To reduce the grid orientation effect, a nine-point, finite-difference reservoir simulator is applied to calculate the steam drive processes, and the AMR technique is also contained in the simulator.2D simulations of SAGD process and steam drive good accuracy, and the speed of the calculation is highly faster than the fine grid solutions.
Before applying the AMR algorithm to the complex faulted reservoirs, we should divide structured grids first. Faults in reservoir are treated as porous media with ultra-low absolute permeability rather than internal boundary, and we could treat the faulted reservoirs as heterogeneous reservoirs. A simple method is given to calculate the permeability of the fine grids crossed by faults, and the permeability is calculated by the fluid flow channel rather than set very low values directly. When both grids are crossed by the same fault, we must consider the continuity of the fault while calculating the flow between the two grids. In the refinement criteria for deciding if the regridding is necessary, the local spatial variations of temperature and phase saturations are used as control values. As a result, coarse grids are applied around faults in the simulations until the steam front arrived, and the renormalization method is adopted to calculate the equivalent permeability of coarse grids in the AMR multi-level grid system.2D SAGD and steam drive examples show that treatment methods of the faults won't destroy the physical properties of the faults, and the AMR technique can still have a highly fast speed. Coarse girds used in faulted regions do not reduce the accuracy, and they are a very important part of reasons of the high calculating speed.
When the AMR technique is applied to the numerical simulation of the steam injection process in complex boundary reservoir, there is a difficulty to divide structured grid system. A large rectangular area is used to cover the entire reservoir, and we provide structured grids to this area. The grids outside the boundary will not be involved in the calculation, so this method do not need extra memory. The permeability of grid interface is involved in the calculation instead of the grid permeability to consider the impact of the border comprehensively. Before implementing the AMR algorithm, the parameters on the fine cells on the reservoir boundary are pre-calculated, and a simple and effective pretreatment method is proposed for the grids across the impermeable layer and the porous media. Single-phase flow examples show that the computing solution of the equivalent permeability is very close to the converged solution according to the pretreatment method even the grid size is large. In the refinement criteria for deciding if the regridding is necessary, the local spatial variations of temperature and phase saturations are used as control values. As a result, the boundary area adopts the coarse grids automatically before the temperature and saturations fronts reach. The same pretreatment method is utilized to calculate the equivalent permeability of the coarse cells of different levels of AMR grids on the reservoir boundary.2D steam drive process with horizontal wells and SAGD process simulation show that the proposed AMR technique is fast with good accuracy. Adoption of the coarse grids rather than the fine grids in the boundary area, according to the AMR refinement criteria, does not reduce the accuracy of the simulations. The same method is applied to the3D simulation of groundwater heat pump system, and the AMR technique is still very successful.
In summary, the object of this paper is to apply the AMR algorithm to the numerical simulation of steam thermal recovery process in complex petroleum reservoirs, and problems relevant to the AMR simulation are studied. The numerical results demonstrate that the AMR technique has a huge advantage in the calculation efficiency, and the accuracy is good compared with the fine grid solutions. We expect the AMR technique to have a practical help to the complex petroleum reservoir simulations in the future.
本文工作的目标是将自适应网格法应用于复杂地层稠油油藏蒸汽注采过程的数值模拟,并研究引入自适应网格法后遇到的相关数学物理问题。
对均质油藏的自适应网格展开研究。利用结构化程序设计方案编写蒸汽注采自适应网格法程序,结构化的程序能够很方便地将后续研究工作引入,从而大幅度缩减数值程序的编写时间。二维与三维的均质油藏蒸汽辅助重力泄油(SAGD)算例显示,自适应网格法能大幅度提高模拟的计算速度,并且保持很高的计算精度。自适应网格结构也显示计算区域中的精细网格数量大幅度减少。
在将自适应网格技术应用到非均质油藏的模拟中时,对于自适应网格系统中的各层次粗网格,采用重正化方法计算等效渗透率。为削减网格取向效应的影响,对二维蒸汽驱的模拟采用九点离散格式,并建立自适应网格系统下非均质油藏的控制方程的离散方法。二维非均质油藏SAGD算例和蒸汽驱算例显示,在此类油藏的模拟中,自适应网格法也能保持很高的计算精度,并且能大幅度提高计算速度。
对于复杂断层油藏的模拟,本文提出将断层处理成渗透率极低的介质,从而将断层油藏类比于非均质油藏,对整个断层油藏划分均匀的结构化网格。对于网格系统中被断层穿过的网格,文中对其等效渗透率的处理提出按流体流通通道进行计算的方法,而不是直接将网格的等效渗透率置为极低值。对于同时被断层穿过的相邻网格,本文提出,在计算它们之间的流量时,必须考虑断层连续性的影响。对断层油藏划分结构化网格后,将自适应网格技术应用其中。自适应网格系统中网格粗化仅依据所选物理量的空间变化,不专门针对断层附近区域划分精细网格。断层区域将依据其附近物理量的变化适时地采用较粗网格计算,从而进一步提高自适应网格法程序的计算速度。对于断层区域附近各层次粗网格,采用重正化方法计算其等效渗透率。二维SAGD算例及蒸汽驱算例表明,将断层处理成渗透介质没有破坏断层本身的物理性质,而自适应网格法的引入大幅度提升了断层油藏模拟的计算速度,同时与全精细网格法相比,自适应网格法也保持了很高的计算精度。
在对复杂边界油藏进行数值研究时,结构化网格往往难以划分。本文提出以一块较大的矩形区域覆盖整个计算区域,针对该矩形区域划分结构化网格的办法。网格结构中,位于边界外的网格并不参与计算,整个计算复杂度没有增加。为充分考虑边界的影响,本文采用界面渗透率代替网格渗透率进行数值计算的方案,对被边界穿过的界面提出渗透率的预处理方法。单相流的验证算例显示,该方案与预处理方法对于复杂边界的模拟能够取得很高的计算精度,即使在网格尺度较大的情况下,计算所得的解依然很接近收敛解。将自适应网格技术应用到复杂边界油藏的模拟中时,网格系统的粗化判据仅为所选物理量的空间变化,并不考虑网格是否位于边界。在计算过程中,计算边界附近会适时的采用粗网格,从而大幅度提高了计算速度。对于边界处的各层次粗网格,采用和精细网格同样的预处理方法计算其界面等效渗透率。二维水平井蒸汽驱算例以及SAGD算例显示,自适应网格法能保持很高计算速度,边界处适时地采用粗网格计算并不影响自适应网格法的计算精度。将文中方案与预处理方法推广到三维问题的计算,三维复杂地层地下水源热泵问题的模拟也显示出AMR计算效率与计算精度上的优势。
综上,本文的工作是对复杂地层稠油油藏蒸汽注采数值模拟自适应网格算法及实施自适应网格算法时遇到的数学物理问题展开研究。文中的数值算例显示自适应网格算法在计算效率上有着巨大的优势,同时在计算精度上与全精细网格法非常吻合。希望自适应网格法能够对生产现场的复杂地层油藏蒸汽注采数值模拟带来帮助。
Steam injection is commonly used to recover heavy oils from petroleum reservoirs. Numerical simulation is a very difficult task, since high nonlinearity in the problem formulation may lead to very sharp thermal and saturation fronts. Because of these rapid variations of the physical quantities across the fronts, numerical grids must be fine enough to achieve reasonable accuracy. As a result, huge CPU time and memory are needed if applying a uniform fine grid to the whole calculation area. In addition, due to the complexity of the reservoir, unstructured meshes are widely used, which is bound to increase the complexity of the numerical discrete. If we use structured grids, the scale should be fine enough to satisfy the simulation accuracy of the complex region. Therefore, improving the calculation efficiency for such problems has great theoretical and practical interest, and the adaptive mesh refinement(AMR) technique is a class of strategies that address this problem.
In this paper, we present an application of the AMR technique to the numerical simulation of steam thermal recovery processes in complex petroleum reservoirs. And several problems relevant to the AMR numerical simulations are carefully studied.
For homogeneous reservoirs, we suggest using structured program design to write the AMR programs. We can easily check the flow-up studies and the preparation time of the numerical procedure can be substantially reduced. The numerical examples for2D and3D SAGD process shows that, the AMR technique results are fast compared with the solutions under referenced uniformly fine grids, and can give good accuracy. AMR grid structure shows that the number of fine grids of the reservoir is significantly reduced.
For heterogeneous reservoirs, effective permeability of the coarse grids must be calculated while we apply the AMR techniques, and the renormalization method is adopted. To reduce the grid orientation effect, a nine-point, finite-difference reservoir simulator is applied to calculate the steam drive processes, and the AMR technique is also contained in the simulator.2D simulations of SAGD process and steam drive good accuracy, and the speed of the calculation is highly faster than the fine grid solutions.
Before applying the AMR algorithm to the complex faulted reservoirs, we should divide structured grids first. Faults in reservoir are treated as porous media with ultra-low absolute permeability rather than internal boundary, and we could treat the faulted reservoirs as heterogeneous reservoirs. A simple method is given to calculate the permeability of the fine grids crossed by faults, and the permeability is calculated by the fluid flow channel rather than set very low values directly. When both grids are crossed by the same fault, we must consider the continuity of the fault while calculating the flow between the two grids. In the refinement criteria for deciding if the regridding is necessary, the local spatial variations of temperature and phase saturations are used as control values. As a result, coarse grids are applied around faults in the simulations until the steam front arrived, and the renormalization method is adopted to calculate the equivalent permeability of coarse grids in the AMR multi-level grid system.2D SAGD and steam drive examples show that treatment methods of the faults won't destroy the physical properties of the faults, and the AMR technique can still have a highly fast speed. Coarse girds used in faulted regions do not reduce the accuracy, and they are a very important part of reasons of the high calculating speed.
When the AMR technique is applied to the numerical simulation of the steam injection process in complex boundary reservoir, there is a difficulty to divide structured grid system. A large rectangular area is used to cover the entire reservoir, and we provide structured grids to this area. The grids outside the boundary will not be involved in the calculation, so this method do not need extra memory. The permeability of grid interface is involved in the calculation instead of the grid permeability to consider the impact of the border comprehensively. Before implementing the AMR algorithm, the parameters on the fine cells on the reservoir boundary are pre-calculated, and a simple and effective pretreatment method is proposed for the grids across the impermeable layer and the porous media. Single-phase flow examples show that the computing solution of the equivalent permeability is very close to the converged solution according to the pretreatment method even the grid size is large. In the refinement criteria for deciding if the regridding is necessary, the local spatial variations of temperature and phase saturations are used as control values. As a result, the boundary area adopts the coarse grids automatically before the temperature and saturations fronts reach. The same pretreatment method is utilized to calculate the equivalent permeability of the coarse cells of different levels of AMR grids on the reservoir boundary.2D steam drive process with horizontal wells and SAGD process simulation show that the proposed AMR technique is fast with good accuracy. Adoption of the coarse grids rather than the fine grids in the boundary area, according to the AMR refinement criteria, does not reduce the accuracy of the simulations. The same method is applied to the3D simulation of groundwater heat pump system, and the AMR technique is still very successful.
In summary, the object of this paper is to apply the AMR algorithm to the numerical simulation of steam thermal recovery process in complex petroleum reservoirs, and problems relevant to the AMR simulation are studied. The numerical results demonstrate that the AMR technique has a huge advantage in the calculation efficiency, and the accuracy is good compared with the fine grid solutions. We expect the AMR technique to have a practical help to the complex petroleum reservoir simulations in the future.
引文
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Friedel H, Grauer R, Marliani C.1997. Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flows[J]. J. Comput. Phys.,134:190-198.
Fulton SR.1997. A Comparison of Multilevel Adaptive Methods for Hurricane Track Prediction[J]. Electronic Trans, on Numer. Anal.,6:120-132.
Fung L S K, Buchanan L, Sharma R.1994. Hybrid-CVFE method for flexible-grid reservoir simulation[J]. SPE Reservoir Engineering,13:188-194.
Fung L S K, Hiebert A D.1992. Reservoir simulation with acontrol-volume finite-element method. SPE Reservoir Engineering,11:349-357.
Gautier Y, Noetinger B.1997. Preferential Flow-Paths Detection for Heterogeneous Reservoirs Using a New Renormalization Technique[J], Trans. Porous Media,26:1-23.
Gerritsen M, Kovscek A, Castanier L, Nilsson J, Younis R, He B.2004. Experimental Investigation and High Resolution Simulator of in-situ Combustion Processes:1. Simulator Design and Improved Combustion with Metallic Additives [J], SPE 86962.
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Haldenwang P, Pignol D.2002. Dynamically adapted mesh refinement for combustion front tracking [J]. Computers and Fluids,31:589-606.
Heinrichs B.1987. Finite difference methods on irregular networks[M]. Brikhauser Verlag.
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Hou TY, Wu XH.1997. A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media [J], J. Comput Phys.,134:169-189.
Howell LH, Pember RB, Colella P.1999. A Conservative Adaptive-Mesh Algorithm for Unsteady, Combined-Mode Heat Transfer Using the Discrete Ordinates Method[J], Numer. Heat Transfer B,35:407-433.
Isea A.1987. Geological Synthesis of the Orinoco oil belt[J]. Eastern Venezuela Journal of Petroleum Geology,10:135-148.
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