A NUMERICAL SIMULATION APPROACH FOR EMBEDDED DISCRETE FRACTURE MODEL COUPLED GREEN ELEMENT METHOD BASED ON TWO SETS OF NODES
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摘要: 对于原始嵌入式离散裂缝模型(EDFM), 在计算包含裂缝单元的基质网格内的压力分布时采用了线性分布假设, 这导致了油藏开发早期对非稳态窜流量的计算精度不足. 因此, 本文提出了一种两套节点格林元法的EDFM数值模拟方法. 两套节点格林元法的核心思想是将压力节点与流量节点区分开, 一套压力节点设置在单元顶点, 另一套流量节点设置在网格边的中点, 满足局部物质守恒、具有二阶精度的同时, 可适用于任意网格类型. 本文将两套节点格林元法与EDFM耦合, 采用了非稳态渗流控制方程的边界积分形式推导了基质网格与裂缝网格之间传质量的新格式, 代替了线性分布假设以提高模拟精度; 此外, 修正后的EDFM能适应任意形态的基质网格剖分, 拓展了原始EDFM仅适用于矩形基质网格、难以考虑复杂油藏边界的局限性. 研究表明: 通过对比商业模拟软件tNavigator® LGR模块与原始EDFM, 验证了本文模型具有较高的早期计算精度; 以复杂油藏边界−缝网−SRV分区模型为例, 通过对比SFEM-COMSOL商业模拟软件, 验证了本文模型处理复杂问题的适应性. 本文研究可用于裂缝性油藏开发动态的精确模拟.Abstract: For the original embedded discrete fracture model (EDFM), the linear-distribution assumption is adopted in calculating the pressure distribution in matrix grids containing fracture elements, which leads to the lack of accuracy in solving unsteady interflux in the early stage of oil reservoir development. Therefore, this paper proposes a numerical simulation approach for EDFM coupled Green element method based on two sets of nodes. The main idea of the Green element method with two sets of nodes is to distinguish pressure nodes from flux nodes, in which one set of pressure nodes is set at the vertex of grids and another set of flux nodes is set at the edge-midpoint of grids. It not only meets the local material conservation and has second-order accuracy, but also can be applied to any grid type. In this paper, the Green element method based on two sets of nodes is coupled with EDFM, and a new scheme of mass transfer between matrix cell and fracture elements is derived by adopting the boundary integral form of the unsteady flow control equation, which replaces the linear distribution assumption to improve the simulation accuracy. In addition, the modified EDFM adapts to any form of matrix mesh generation, which extends the limitations of the original EDFM which is only suitable for rectangular matrix mesh and difficult to consider complex reservoir boundaries. The research shows that the proposed model has high accuracy in the early stage and it is verified by the LGR module of commercial software tNavigator® and the original EDFM. Taking the SRV-zoning model considering fracture networks and complex reservoir boundaries as an example, the flexibility of the proposed model for solving complicated problems is demonstrated by comparing the business simulation software named SFEM-COMSOL. This study can be used for the accurate simulation of dynamic production performance in fractured oil reservoirs.
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表 1 数值算例的输入参数
Table 1. Input parameters of numerical case
Properties Value Properties Value reservoir thickness/m 10 initial reservoir pressure/MPa 20 matrix porosity 0.15 matrix permeability/mD 1 rock compressibility coefficient/MPa−1 1.08 × 10−4 fracture aperture/m 0.005 fracture porosity 0.45 fracture permeability/mD 70000 oil viscosity/(mPa·s) 2 oil compressibility coefficient/MPa−1 3 × 10−3 -
[1] 曹仁义, 程林松, 杜旭林等. 致密油藏渗流规律及数学模型研究进展. 西南石油大学学报(自然科学版), 2021, 43(5): 113-136 (Cao Renyi, Cheng Linsong, Du Xulin, et al. Research progress on fluids flow mechanism and mathematical model in tight oil reservoirs. Journal of Southwest Petroleum University (Science &Technology Edition) , 2021, 43(5): 113-136 (in Chinese) [2] Obembe AD, Hossain ME. A new pseudosteady triple-porosity model for naturally fractured shale gas reservoir//SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, 2015 [3] 苏皓, 雷征东, 李俊超等. 储集层多尺度裂缝高效数值模拟模型. 石油学报, 2019, 40(5): 587-593, 634 (Su Hao, Lei Zhendong, Li Junchao, et al. An effective numerical simulation model of multi-scale fractures in reservoir. Acta Petrolei Sinica, 2019, 40(5): 587-593, 634 (in Chinese) [4] Rao X, Cheng L, Cao R, et al. A modified projection-based embedded discrete fracture model (pEDFM) for practical and accurate numerical simulation of fractured reservoir. Journal of Petroleum Science and Engineering, 2020, 187: 106852 [5] 杜旭林, 程林松, 牛烺昱等. 基于XFEM-MBEM 的嵌入式离散裂缝模型流固耦合数值模拟方法. 力学学报, 2021, 53(12): 3416-3427 (Du Xulin, Cheng Linsong, Niu Langyu, et al. Numerical simulation for coupling flow and geomechanics in embedded discrete fracture model based on XFEM-MBEM. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(12): 3416-3427 (in Chinese) [6] Li L, Lee SH. Efficient field-scale simulation of black oil in a naturally fractured reservoir through discrete fracture networks and homogenized media. SPE Reservoir Evaluation & Engineering, 2008, 11(4): 750-758 [7] Lee SH, Lough MF, Jensen CL. Hierarchical modeling of flow in naturally fractured formations with multiple length scales. Water Resources Research, 2001, 37(3): 443-455 [8] Hajibeygi H, Karvounis D, Jenny P. A hierarchical fracture model for the iterative multiscale finite volume method. Journal of Computational Physics, 2011, 230(24): 8729-8743 [9] 朱大伟, 胡永乐, 崔明月等. 局部网格加密与嵌入式离散裂缝模型耦合预测压裂改造井产能. 石油勘探与开发, 2020, 47(2): 341-348 (Zhu Dawei, Hu Yongle, Cui Mingyue, et al. Productivity simulation of hydraulically fractured wells based on hybrid local grid refinement and embedded discrete fracture model. Petroleum Exploration and Development, 2020, 47(2): 341-348 (in Chinese) doi: 10.11698/PED.2020.02.12 [10] Xue X, Yang C, Onishi T, et al. Modeling hydraulically fractured shale wells using the fast-marching method with local grid refinements and an embedded discrete fracture model. SPE Journal, 2019, 24(6): 2590-2608 [11] Liu L, Huang Z, Yao J, et al. An efficient hybrid model for 3D complex fractured vuggy reservoir simulation. SPE Journal, 2020, 25(2): 907-924 [12] Cao Y, Killough JE. An improved boundary element method for modeling fluid flow through fractured porous medium//SPE Reservoir Simulation Conference, Society of Petroleum Engineers, 2017 [13] Taigbenu AE. A more efficient implementation of the boundary element theory//Proc. 5th International Conference on Boundary Element Technology (BETECH 90), Newark, Delaware, 1990 [14] Taigbenu AE. The Green element method. International Journal for Numerical Methods in Engineering, 1995, 38: 2241-2263 doi: 10.1002/nme.1620381307 [15] Taigbenu AE, Onyejekwe OO. Green element simulations of the transient nonlinear unsaturated flow equation. Applied Mathematical Modelling, 1995, 19(11): 675-684 [16] Taigbenu AE, Onyejekwe OO. A mixed Green element formulation for the transient Burgers equation. International Journal for Numerical Methods in Fluids, 1997, 24(6): 563-578 [17] Taigbenu AE, Akpofure E. The Green Element Method. Springer, 1999 [18] Archer R, Horne RN, Onyejekwe OO. Petroleum reservoir engineering applications of the dual reciprocity boundary element method and the Green element method//Proceedings of Twenty First International Conference on Boundary Element Method Southampton, 1999 [19] Archer R. C1 continuous solutions from the Green element method using Overhauser elements. Applied Numerical Mathematics, 2006, 56(2): 222-229 doi: 10.1016/j.apnum.2005.04.001 [20] Pecher R, Harris SD, Knipe RJ, et al. New formulation of the Green element method to maintain its second-order accuracy in 2D/3D. Engineering Analysis with Boundary Elements, 2001, 25(3): 211-219 [21] Lorinczi P, Harris SD, Elliot L. Modified flux-vector–based Green element method for problems in steady-state anisotropic media-Generalisation to triangular elements. Engineering Analysis with Boundary Elements, 2011, 35: 495-498 [22] Lorinczi P, Harris SD, Elliott L. Unsteady flux-vector-based Green element method. Transport in Porous Media, 2011, 87(1): 207-228 [23] Lorinczi P, Harris SD, Elliott L. Modified flux-vector-based Green element method for problems in steady-state anisotropic media. Engineering Analysis with Boundary Elements, 2009, 33(3): 368-387 [24] Lorinczi P, Harris SD, Elliott L. Modelling of highly-heterogeneous media using a flux-vector-based Green element method. Engineering Analysis with Boundary Elements, 2006, 30(10): 818-833 [25] Taigbenu AE. The flux-correct Green element formulation for linear, nonlinear heat transport in heterogeneous media. Engineering Analysis with Boundary Elements, 2008, 32(1): 52-63 [26] Taigbenu AE. Enhancement of the accuracy of the Green element method: Application to potential problems. Engineering Analysis with Boundary Elements, 2012, 36(2): 125-136 [27] 方思冬, 程林松, 饶翔等. 一种改进格林元方法及在渗流问题中的应用. 计算力学学报, 2021, 38(6): 787-795 (Fang Sidong, Cheng Linsong, Rao Xiang, et al. An improved Green element method and its application in seepage problems. Chinese Journal of Computational Mechanics, 2021, 38(6): 787-795 (in Chinese) doi: 10.7511/jslx20200907001 [28] Rao X, Cheng L, Cao R, et al. A mimetic Green element method. Engineering Analysis with Boundary Elements, 2019, 99: 206-221 [29] Rao X, Cheng L, Cao R, et al. A novel Green element method based on two sets of nodes. Engineering Analysis with Boundary Elements, 2018, 91: 124-131 [30] Persson PO, Strang G. A simple mesh generator in MATLAB. SIAM Review, 2004, 46(2): 329-345 [31] Persson PO. Mesh generation for implicit geometries. [PhD Thesis]. Cambridge: MIT, 2004 [32] Yan CZ, Guo H, Tang ZC. Three-dimensional continuous-discrete pore-fracture mixed seepage model and hydromechanical coupling model to simulate rock fracture driven by fluid. Journal of Petroleum Science and Engineering, 2022, 215: 110510 [33] Yan CZ, Gao Y, Guo H. A FDEM based 3D discrete mixed seepage model for simulating fluid driven fracturing. Engineering Analysis with Boundary Elements, 2022, 140: 447-463 [34] Jia P, Cheng L, Huang S, et al. Transient behavior of complex fracture networks. Journal of Petroleum Science and Engineering, 2015, 132: 1-17 [35] 贾品, 程林松, 黄世军等. 水平井体积压裂缝网表征及流动耦合模型. 计算物理, 2015, 32(6): 685-692 (Jia Pin, Cheng Linsong, Huang Shijun, et al. Characterization of fracture network by volume fracturing in horizontal wells and coupled flow model. Chinese Journal of Computational Physics, 2015, 32(6): 685-692 doi: 10.3969/j.issn.1001-246X.2015.06.008 [36] Peaceman DW. Interpretation of well-block pressure in numerical reservoir simulation with nonsquare gridblocks and anisotropic permeability. SPE Journal, 1983, 18(3): 183-194 -