﻿ 裂缝性油藏离散裂缝网络模型与数值模拟
 西南石油大学学报(自然科学版)  2017, Vol. 39 Issue (3): 121-127

Discrete Fracture Network Modeling and Numerical Simulation of Fractured Reservoirs
ZHANG Liehui , JIA Ming, ZHANG Ruihan, GUO Jingjing
State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China
Abstract: Discrete fracture network models have become the focus of much research owing to their ability to better characterize the non-homogeneity of fractured reservoirs. The FILTERSIM multi-point geostatistical method was introduced into the discrete fracture network model to overcome the deficiencies of the Monte Carlo method and to provide a fracture network model for the subsequent numerical simulation of the flow. Based on the oil-water two-phase flow model, the finite-element numerical solution of the degree of oil-water two-phase saturation was derived. Taken together with the constructed fracture network model, the dynamic process of the oil displacement by water and pattern of moisture content increase during single-well injections were studied. The study showed that compared with the homogeneous model, injection of water through the channels of the fracture network in the discrete fracture network model led to the premature emergence of water from oil wells. After the emergence of water from the oil well, the two models exhibited consistent patterns of water cut increase.
Key words: discrete fracture network     multi-point geo-statistics     numerical simulation     fractured reservoirs     oil-water twophase flow

1 FILTERSIM基本原理 1.1 FILTERSIM方法原理

FILTERSIM方法主要包含3个部分：模板降维，模板分类和序贯模拟。但在这之前，需要用搜索模板扫描训练图像以构建数据模板数据库。

 $\boldsymbol{t}(u) = \left[t(u + {h_1}), t(u + {h_2}), \cdots, t(u + {h_N})\right]$ (1)

FILTERSIM算法给出一组滤波器fk，（k = 1, 2, ..., F），将每一个滤波器作用在一个数据模板t(u)上，就会得到一个滤波分。这样，一组滤波器作用在一个数据模板上将会产生F个滤波分，表示为S(u)，F = Sk(u)，（k = 1, 2, ..., F），Sk(u)的计算公式为

 ${S_k}(u) = \sum\limits_{j = 1}^N {{f_k}({h_j})} t(u + {h_j}), {\kern 20pt}k = 1, 2, ...F$ (2)

 ${p_{{\rm{rot}}}}({h_j}) = \dfrac{1}{{{c}}}\sum\limits_{i = 1}^{{c}} {t(u_i + {h_j})}, {\kern 10pt}j = 1, 2, \cdots, N$ (3)

 $D(u) = \sum\limits_{d = 1}^3 {\omega (d){\rm{Ave}}\left| {{d_{{\rm{at}}}}(u + {h_j}) - {p_{{\rm{rot}}}}({h_j})} \right|}$ (4)

1.2 FILTERSIM方法建立离散裂缝模型

 图1 裂缝网络训练图像 Fig. 1 The training image of the fracture networks
 图2 25次模拟结果的平均值 Fig. 2 The E-types of 25 simulations
2 离散裂缝油水两相流渗流模型 2.1 基本假设

（1） 油藏中仅仅存在油水两相，且它们的流动均符合达西定律；

（2） 流体在基岩中为二维流动，考虑毛细管压力；裂缝中为一维流动，不考虑毛管力作用；

（3） 基岩与基岩中的流体皆为压缩系数为常数的微可压缩介质，裂缝与裂缝中的流体不可压缩；

（4）不考虑重力作用。

2.2 数学模型

 $-{\phi _{{\rm{m0}}}}{C_{{\rm{tm}}}}\dfrac{{\partial {p_{{\rm{wm}}}}}}{{\partial t}} + \nabla \cdot ({\lambda _{{\rm{om}}}}{p_{{\rm{cm}}}}) + \\ \nabla \cdot ({\lambda _{{\rm{tm}}}}\nabla {p_{{\rm{wm}}}}) + {Q_{{\rm{ovm}}}} +{Q_{{\rm{wvm}}}} - \delta q_{{\rm{mfT}}}^{\rm{*}} = 0$ (5)
 $- \nabla \cdot {\rm{(}}{{\rm{\lambda }}_{{\rm{wm}}}}\nabla {{{p}}_{{\rm{wm}}}}{\rm{)}} + {{\rm{\phi }}_{{\rm{m0}}}}\dfrac{{\partial {{{S}}_{{\rm{wm}}}}}}{{\partial {{t}}}} = {{{Q}}_{{\rm{wvm}}}} - \delta q_{{\rm{mfw}}}^{\rm{*}}$ (6)

 $\dfrac{\partial }{{\partial l}}\left({\lambda _{{\rm{tf}}}}\dfrac{{\partial {p_{{\rm{wf}}}}}}{{\partial l}}\right) + \delta q_{{\rm{mfT}}}^{\rm{*}} = 0$ (7)
 $- \dfrac{\partial }{{\partial l}}\left({\lambda _{{\rm{wf}}}}\dfrac{{\partial {p_{{\rm{wf}}}}}}{{\partial l}}\right) + {\phi _{{\rm{f0}}}}\dfrac{{\partial {S_{{\rm{wf}}}}}}{{\partial t}} = - \delta q_{{\rm{mfw}}}^{\rm{*}}$ (8)

 $\begin{array}{*{20}{c}} {{\lambda _{ij}} = \dfrac{{{K_{{\rm{r}}ij}}{K_j}}}{{{\mu _i}}}}, {\kern 10pt}{(i = {\rm{o, w;}}j = {\rm{m, f}})} \end{array}$ (9)
 $\begin{array}{*{20}{c}} {{\lambda _{{\rm{t}}j}} = {\lambda _{{\rm{o}}j}} + {\lambda _{{\rm{w}}j}}}, {\kern 10pt}{(j = {\rm{m, f}})} \end{array}$ (10)
 ${C_{\rm{t}}} = {C_{{\rm{to}}}}{S_{\rm{o}}} + {C_{{\rm{tw}}}} + {S_{\rm{w}}}$ (11)
 ${{C}_{\text{t}i}}={{C}_{\mathtt{ϕ}}}+{{C}_{i}}\left( i=\text{o},\text{w} \right)$ (12)
 $q_{{\rm{mfT}}}^{\rm{*}} = q_{{\rm{mfo}}}^{\rm{*}} + q_{{\rm{mfw}}}^{\rm{*}}$ (13)
 $\delta = \left\{ \begin{array}{l} \begin{array}{*{20}{c}} 1&{(\mbox{汇源处})} \end{array}\\ \begin{array}{*{20}{c}} 0&{(\mbox{非汇源处})} \end{array} \end{array} \right.$ (14)
2.3 有限元数值求解

 $\left({{\boldsymbol{K}}_{\rm{m}}} + {{\boldsymbol{K}}_{\rm{f}}}\right){{\boldsymbol{p}}_{\rm{w}}} + {\boldsymbol{C}}\dfrac{{\partial {{\boldsymbol{p}}_{\rm{w}}}}}{{\partial t}} = {\boldsymbol{F}}$ (15)

 ${{\boldsymbol{K}}_{\rm m}} = \sum {{{\boldsymbol{K}}_{{\rm{m, e}}}}}$
 ${{\boldsymbol{K}}_{\rm{f}}} = \sum {{{\boldsymbol{K}}_{{\rm{f, e}}}}}$
 ${\boldsymbol{C}} = \sum {{{\boldsymbol{C}}_{\rm{e}}}}$
 ${\boldsymbol{F}} = \sum {{{\boldsymbol{F}}_{\rm{e}}}}$
 ${{\boldsymbol{K}}_{{\text{m, e}}}} = \iint\limits_{{{\varOmega} _{\text{m}}}} {\left( {{\lambda _{{\text{tm}}x}}\dfrac{{\partial {\boldsymbol{N}}_{\text{m}}^{\text{T}}}}{{\partial x}}\dfrac{{\partial {{\boldsymbol{N}}_{\text{m}}}}}{{\partial x}} + {\lambda _{{\text{tm}}y}}\dfrac{{\partial {\boldsymbol{N}}_{\text{m}}^{\text{T}}}}{{\partial y}}\dfrac{{\partial {{\boldsymbol{N}}_{\text{m}}}}}{{\partial y}}} \right)}{\text{d}}{{\varOmega} _{\text{m}}}$
 ${{\boldsymbol{K}}_{{\text{f, e}}}} = \int\limits_{{{\varOmega} _l}} {{\lambda _{{\text{tf}}}}\dfrac{{\partial {\boldsymbol{N}}_l^{\text{T}}}}{{\partial l}}\dfrac{{\partial {{\boldsymbol{N}}_l}}}{{\partial l}}} {\text{d}}l$
 ${{\boldsymbol{C}}_{\text{e}}}=\iint\limits_{{{\mathit{\Omega} }_{\text{m}}}}{{{\phi }_{\text{m}0}}}{{\boldsymbol{C}}_{\text{tm}}}\boldsymbol{N}_{\text{m}}^{\text{T}}{{\boldsymbol{N}}_{\text{m}}}\text{d}{{\mathit{\Omega }}_{\text{m}}};$
 ${{\boldsymbol{F}}_{\text{e}}} = - \iint\limits_{{{\varOmega} _{\text{m}}}} {\left( {{\lambda _{{\text{t m}}x}}\dfrac{{\partial {\boldsymbol{N}}_{\text{m}}^{\text{T}}}}{{\partial x}}\dfrac{{\partial {p_{\text{c}}}}}{{\partial x}} + {\lambda _{{\text{tm}}y}}\dfrac{{\partial {\boldsymbol{N}}_{\text{m}}^{\text{T}}}}{{\partial y}}\dfrac{{\partial {p_{\text{c}}}}}{{\partial y}}} \right)}{\text{d}}{{\varOmega} _{\text{m}}} - \\{\kern 40pt}\iint\limits_{{{\varOmega} _{\text{m}}}} {\left( {{Q_{{\text{ovm}}}} + {Q_{{\text{wvm}}}}} \right){\boldsymbol{N}}_{\text{m}}^{\text{T}}{\text{d}}{{\varOmega} _{\text{m}}}}$

 $\left( {{{\boldsymbol{K}}_{\text{m}}} + {{\boldsymbol{K}}_{\text{f}}}} \right){\boldsymbol{p}}_{\text{w}}^{(n + 1)} + {\boldsymbol{C}}\dfrac{{{\boldsymbol{p}}_{\text{w}}^{(n + 1)} - {\boldsymbol{p}}_{\text{w}}^{(n)}}}{{\Delta t}} = {\boldsymbol{F}}$ (16)

 $S_{{\text{wm}}}^{(n + 1)} = S_{{\text{wm}}}^{(n)} + \frac{{\Delta t}}{{{\phi _{{\text{m0}}}}}}\left[ {{Q_{{\text{wvm}}}} - \delta q_{{\text{mfw}}}^{\text{*}} + \nabla \cdot \left( {{\lambda _{{\text{wm}}}}\nabla {p_{{\text{wm}}}}} \right)} \right]$ (17)
 $S_{{\text{wf}}}^{(n + 1)} = S_{{\text{wf}}}^{(n)} + \dfrac{{\Delta t}}{{{\phi _{{\text{f0}}}}}}(\dfrac{\partial }{{\partial l}}({\lambda _{{\text{wf}}}}\dfrac{{\partial {p_{{\text{wf}}}}}}{{\partial l}}) - \delta q_{{\text{mfw}}}^{\text{*}})$ (18)
3 实例分析

 图3 图像处理后的离散裂缝网络模型 Fig. 3 Discrete fracture networks model after image processing
 图4 网格剖分 Fig. 4 Mesh generation

 图5 采油井含水率曲线图 Fig. 5 The water cut of production well

 图6 离散裂缝模型和均质模型中的油相饱和度分布图 Fig. 6 Oil phase saturation distribution in discrete fracture model and homogeneity model
4 结论

（1） 将广泛运用于沉积相建模的多点地质统计学方法引入到离散裂缝网络建模中，能够较好地再现训练图像中裂缝分布信息，为油藏模拟提供了较可靠的裂缝网络模型。

（2） 耦合离散裂缝网络建模与有限元数值模拟，能够较好地模拟复杂裂缝网络非均质储层水驱动态。

NDT-搜索模板像素；

NTI-训练模板像素；

t(u)-中心点为u的数据模板；

u + hj-中心点为u的邻近点；

N-数据模板维数；

fk-第k个滤波器；

F-滤波器数目；

Sk(u)-第k个滤波分；

prot(hj)-数据原型；

c-某一类中的数据模板个数；

dat(u)-中心点为u的数据事件；

D(u)-数据事件与数据原型的距离；

ω(d)-第d类条件数据的权值；

Ave-求平均值函数；

φm0-初始基质孔隙度，%；

Ctm-基质总压缩系数，MPa-1

pwm-基质水相压力，MPa；

t-时间，s；

λom-基质油相流度，D/(mPa·s)；

pcm-基质毛细管压力，MPa；

λtm-基质总流度，D/(mPa·s)；

Qovm-基质油相汇源，s-1

Qwvm-基质水相汇源，s-1

δ-δ函数；

qmfT*-裂缝和基质之间总的窜流量，s-1

λwm-基质水相流度，D/(mPa·s)；

Swm-基质水相饱和度，无因次；

qmfw*-裂缝和基质之间水相窜流量，s-1

λtf-裂缝总流度，D/(mPa·s)；

pwf-裂缝水相压力，MPa；

l-沿裂缝走向的局部坐标，m；

λwf-裂缝水相流度，D/(mPa·s)；

φf0-初始裂缝孔隙度，%；

Swf-裂缝水相饱和度，无因次；

Kr-相对渗透率，%；

K-渗透率，D；

Ct-油水两相总压缩系数，MPa-1

Cto-油相总压缩系数，MPa-1

So-油相饱和度，%；

Ctw-水相总压缩系数，MPa-1

Sw-水相饱和度，%；

Cφ-孔隙度压缩系数，MPa-1

Co-油相压缩系数，MPa-1

Cw-水相压缩系数，MPa-1

qmfo*-裂缝和基质之间油相窜流量，s-1

Km-基质渗透率总体矩阵；

Kf-裂缝渗透率总体矩阵；

pw-压力总体矩阵；

C-压缩系数总体矩阵；

F-外力总体矩阵；

Km, e-基质渗透率单位矩阵；

Kf, e-裂缝渗透率单位矩阵；

Ce-压缩系数单位矩阵；

Fe-外力单位矩阵；

Ωm-基质内求解单位渗透率矩阵积分区域；

x, y-积分区域坐标系；

λtmx-x方向基质总流度，D/（mPa·s）；

λtmy-y方向基质总流度，D/（mPa·s）；

Nm-基质中形状函数矩阵；

Nl-裂缝中形状函数矩阵；

Ctm-基质中油水两相总压缩系数，MPa-1

pc-毛细管压力，MPa；

pw(n+1)-n + 1时间步时的压力矩阵；

Δt-时间步长，s；

Swm(n+1)-n + 1时间步时的基质水相饱和度，%；

Swm(n)-n时间步时的基质水相饱和度，%；

Swf(n+1)-n + 1时间步时的裂缝水相饱和度，%；

Swf(n)-n时间步时的裂缝水相饱和度，%。

 [1] BEYDOUN ZIAD R. Arabian plate oil and gas: Why so rich and so prolific?[J]. Episodes-Newsmagazine of the International Union of Geological Sciences, 1998, 21(2): 74–81. [2] DERSHOWITZ W S, LA POINTE P R, EIBEN T, et al. Integration of discrete fracture network methods with conventional simulator approaches[C]. SPE 62498, 2000. doi: 10.2118/62498-PA [3] KIM J G, DEO M D. Comparison of the performance of a discrete fracture multiphase model with those using conventional methods[C]. SPE 51928, 1999. doi: 10.2118/-51928-MS [4] OUENES A, HARTLEY L J. Integrated fractured reservoir modeling using both discrete and continuum approaches[C]. SPE 62939, 2000. doi: 10.2118/62939-MS [5] BEACHER G B, LANNEY N A, EINSTEIN H H. Statistical description of rock properties and sampling[C]. SPE 77-0400, 1977. [6] TRAN N H, CHEN Z, RAHMAN S S. Integrated conditional global optimization for discrete fracture network modeling[J]. Computers & Geosciences, 2006, 32: 17–27. doi: 10.1016/j.cageo.2005.03.019 [7] XU C, DOWD P. A new computer code for discrete fracture network modeling[J]. Computers & Geosciences, 2010, 36: 292–301. doi: 10.1016/j.cageo.2009.05.012 [8] 宋晓晨, 徐卫亚. 裂隙岩体渗流模拟的三维离散裂隙网络数值模型(Ⅰ):裂隙网络的随机生成[J]. 岩石力学与工程学报, 2004, 23(12): 2015–2020. SONG Xiaochen, XU Weiya. Numerical model of threedimensional discrete fracture network for seepage in fractured rocks(Ⅰ): generation of fracture network[J]. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(12): 2015–2020. doi: 10.3321/j.issn:1000-6915.2004.-12.012 [9] 郑松青, 张宏方, 刘中春, 等. 裂缝性油藏离散裂缝网络模型[J]. 大庆石油学院学报, 2011, 35(6): 49–54. ZHENG Songqing, ZHANG Hongfang, LIU Zhongchun, et al. Discrete fracture network model for fractured reservoirs[J]. Journal of Daqing Petroleum Institute, 2011, 35(6): 49–54. doi: 10.3969/j.issn.2095-4107.2011.06.009 [10] 杨坚, 呂心瑞, 李江龙, 等. 裂缝性油藏离散裂缝网络随机生成及数值模拟[J]. 油气地质与采收率, 2011, 18(6): 74–77. YANG Jian, LÜ Xinrui, LI Jianglong, et al. Study on discrete fracture network random generation and numerical simulation of fractured reservoir[J]. Petroleum Geology and Recovery Efficiency, 2011, 18(6): 74–77. doi: 10.-3969/j.issn.1009-9603.2011.06.019 [11] ZHANG T F. Filter-based training pattern classification for spatial pattern simulation[D]. Stanford: Stanford University, 2006. [12] 姚军, 王子胜, 张允, 等. 天然裂缝性油藏的离散裂缝网络数值模拟方法[J]. 石油学报, 2010, 31(2): 284–288. YAO Jun, WANG Zisheng, ZHANG Yun, et al. Numerical simulation method of discrete fracture network for naturally fractured reservoirs[J]. Acta Petrolei Sinica, 2010, 31(2): 284–288. doi: 10.7623/syxb201002018 [13] 张芮菡. 基于有限元有限体积法的裂缝性油藏数值模拟研究[D]. 成都: 西南石油大学, 2014. ZHANG Ruihan. Study on fractured reservoir numerical simulation based on infinite element-infinite volume method[D]. Chengdu: Southwest Petroleum University, 2014. [14] 吕心瑞, 姚军, 黄朝琴, 等. 基于有限体积法的离散裂缝模型两相流动模拟[J]. 西南石油大学学报(自然科学版), 2012, 34(6): 123–130. LÜ Xinrui, YAO Jun, HUANG Zhaoqin, et al. Study on discrete fracture model two-phase flow simulation based on finite volume method[J]. Journal of Southwest Petroleum University (Science & Technology Edition), 2012, 34(6): 123–130. doi: 10.3863/j.issn.1674-5086.2012.06.-018 [15] 孟欣然, 梁堰波, 孟宪海, 等. 基于FILTERSIM算法的油藏沉积微相模拟研究[J]. 计算机工程设计, 2013, 34(2): 545–549. MENG Xinran, LIANG Yanbo, MENG Xianhai, et al. Using FILTERSIM for reservoir faces simulation[J]. Computer Engineering and Design, 2013, 34(2): 545–549. doi: 10.16208/j.issn1000-7024.2013.02.041 [16] 尹艳树, 张昌民, 李玖勇, 等. 多点地质统计学研究进展与展望[J]. 古地理学报, 2011, 13(2): 245–252. YIN Yanshu, ZHANG Changmin, LI Jiuyong, et al. Progress and prospect of multiple-point geostatistics[J]. Journal of Palaeogrography, 2011, 13(2): 245–252. doi: 10.7605/gdlxb.2011.02.013 [17] Ahmadi R, Masihi M, Rasaei M R, et al. A sensitivity study of FILTERSIM algorithm when applied to DFN modeling[J]. J Petrol Explor Prod Technol, 2014, 4(2): 153–174. doi: 10.1007/s13202-014-0107-0 [18] 周金应, 桂碧雯, 林闻. 多点地质统计学在滨海相储层建模中的应用[J]. 西南石油大学学报(自然科学版), 2010, 32(6): 70–73. ZHOU Jinying, GUI Biwen, LIN Wen. Application of multi-point geostatistics in offshore reservoir modeling[J]. Journal of Southwest Petroleum University (Science & Technology Edition), 2010, 32(6): 70–73. doi: 10.3863/j.issn.1674-5086.2010.06.014 [19] 吴小军, 李晓梅, 谢丹, 等. 多点地质统计学方法在冲积扇构型建模中的应用[J]. 岩性油气藏, 2015, 27(5): 87–91. WU Xiaojun, LI Xiaomei, XIE Dan, et al. Application of multi-point geostatistics method to structure modeling of alluvial fan[J]. Lithologic Reservoirs, 2015, 27(5): 87–91. doi: 10.3969/j.issn.1673-8926.2015.05.015 [20] ZHANG T, DU Y, HUANG T, et al. Stochastic simulation of patterns using ISOMAP for dimensionality reduction of training images[J]. Computers & Geosciences, 2015, 79: 82–93. doi: 10.1016/j.cageo.2015.03.010 [21] WU J B, BOUCHER A, ZHANG T F. A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM[J]. Computers & Geosciences, 2008, 34(12): 1863–1876. doi: 10.1016/j.cageo.2007.08.008