﻿ 考虑次生裂缝的页岩气藏有限导流缝网模型
 西南石油大学学报(自然科学版)  2019, Vol. 41 Issue (6): 139-145

1. 油气藏地质及开发工程国家重点实验室·西南石油大学, 四川 成都 610500;
2. 西南交通大学利兹学院, 四川 成都 611756

The Finite-conductivity Fracture Networks Model in Shale Gas Reservoirs with Consideration of Induced Fractures
FANG Quantang1 , LI Zhenglan1, DUAN Yonggang1, WEI Mingqiang1, ZHANG Yuyi2
1. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China;
2. Southwest Jiaotong University Leeds Joint School, Chengdu, Sichuan 611756, China
Abstract: In order to analyze the influence of secondary induced fractures on the pressure response of shale gas wells, a finite conductivity fracture network flow model of shale gas reservoir with coupling multiple migration mechanism has been established, and the pressure dynamic characteristics have been studied. Firstly, the analytical solution of pressure in shale gas reservoir has been obtained by employing Laplace space source function, local coordinate transformation and superposition principle. Then, based on the finite difference method and the flow distribution transformation of intersection element, the numerical solution of fracture element has been derived. By coupling the flow of gas reservoir and fracture, the pressure response curve considering the influence of secondary fracture was drawn, and the influences of characteristic parameters (such as the number of secondary fracture groups, secondary fracture angle, and fracture conductivity, etc.) were analyzed. The results show that there are 10 typical flow stages, which can effectively characterize the influence of secondary fractures. In addition, the presented model will be helpful for understanding the transient performance of multi-stage fractured horizontal wells with consideration of induced fractures.
Keywords: shale gas reservoir     secondary induced fractures     fractured horizontal well     finite-conductivity     pressure performance

1 物理模型

(1) 页岩气藏为双重介质气藏，气体在基质中以拟稳态形式流动、天然裂缝中以渗流形式流动，气体解吸满足Langmuir单分子层等温吸附定律[21]。(2)人工主裂缝与井筒正交等间距分布，多组次生裂缝与主缝成一定角度相交，如图 1所示。主缝与次缝均为有限导流裂缝。(3)气井以恒定产量进行生产，井筒中不存在压降，具有无限导流能力。(4)不考虑重力和页岩储层的压缩性。

 图1 物理模型示意图 Fig. 1 Schematic of interlaced fracture networks
2 数学模型

2.1 页岩气藏裂缝渗流-基质拟稳态扩散的双重介质模型

 $\dfrac{1}{{r_{\rm{D}}^2}}\dfrac{\partial }{{\partial {r_{\rm{D}}}}}\left( {r_{\rm{D}}^2\dfrac{{\partial {{\overline \psi }_{{\rm{fD}}}}}}{{\partial {r_{\rm{D}}}}}} \right) = f\left( u \right){\overline \psi _{{\rm{fD}}}}$ (1)

$r_{\rm{D}}$—球形基质径向距离，无因次；

$\overline \psi _{{\rm{fD}}}$— Laplace空间拟压力，无因次；

$f(u)$—流动方程系数，无因次；

$f\left(u \right) = \omega u + \dfrac{{\alpha u\lambda \left({1 - \omega } \right)}}{{u + \lambda }}$

$\omega$—储容比，无因次；

$u$— Laplace变量；

$\lambda$—窜流系数，无因次；

$\alpha$—吸附解吸常数，无因次；

$\alpha = \dfrac{{{V_{\rm{L}}}{p_{\rm{L}}}}}{{\left({{p_{\rm{L}}} + {p_{\rm{f}}}} \right)\left({{p_{\rm{L}}} + {p_{\rm{i}}}} \right)}}\dfrac{{{p_{{\rm{sc}}}}{q_{{\rm{sc}}}}T}}{{{\rm{ \mathsf{ π} }}K_{\rm{f}}h{T_{{\rm{sc}}}}}}\dfrac{{{\mu _{\rm{i}}}{Z_{\rm{i}}}}}{{2{p_{\rm{i}}}}}$

$V$—基质中气体平均浓度，sm$^3$/m$^3$

$p$—压力，MPa；

$q$—页岩气产量，m$^3$/s；

$T$—温度，K；

$\mu$—气体黏度，Pa$\cdot$s；

$K_{\rm{f}}$—裂缝渗透率，m$^2$

$h$—气藏厚度，m；

$Z$—气体偏差因子，无因次。

f—天然裂缝；

D—无因次量；

sc—地面状态；

i—初始状态；

L— langmuir常量。

 ${\rm{\Delta }}{\overline \psi _{\rm{f}}} = \dfrac{{{p_{{\rm{sc}}}}T}}{{{T_{{\rm{sc}}}}}}\dfrac{{\overline q (u)}}{{{\rm{ \mathsf{ π} }}{K_{\rm{f}}}L{h_{\rm{D}}}}}{{{K}}_0}\left( {{R_{\rm{D}}}\sqrt {f\left( u \right)} } \right)$ (2)

$\overline q (u)$— Laplace空间下点源流量，m$^3$/s；

$L$—参考长度，m；

$K_0$()—零阶贝塞尔函数；

$R_{\rm{D}}$—源点到场点的距离，无因次。

 $\overline \psi _{{\rm{fD}}}^{}\left( {{x_{{\rm{D}}p, q}}, {y_{{\rm{D}}p, q}}} \right) = \sum\limits_i {\sum\limits_j {} } {\overline q _{{\rm{D}}i, j}}\left( u \right)\int\limits_{} {{{{K}}_0}} \\ \left[ {\sqrt {f\left( u \right)} \sqrt {{{\left( {{x_{{\rm{D}}p, q}} - {x_{{\rm{wD}}}}} \right)}^2} + {{\left( {{y_{{\rm{D}}p, q}} - {x_{{\rm{wD}}}}\tan \theta } \right)}^2}} } \right]\sqrt {1 + {{\tan }^2}\theta } {\rm{d}}{x_{{\rm{wD}}}}$ (3)

$\left({{x_{{\rm{D}}p, q}}, {y_{{\rm{D}}p, q}}} \right)$—局部坐标系下场点离散段($p, q$)坐标；

$x_{{\rm{wD}}}$—无因次源点$x$坐标；

$p$$q—场点主缝编号和次缝编号； i$$j$—源点主缝编号和次缝编号；

$\theta$—主缝与次缝夹角，(°)。

2.2 有限导流裂缝流动模型

 $\dfrac{{{\partial ^2}{{\overline \psi }_{{\rm{flD}}}}}}{{\partial l_{\rm{D}}^2}} - \dfrac{{2{\rm{ \mathsf{ π} }}}}{{{R_{{\rm{fD}}}}}}{\overline q _{{\rm{flD}}}}\left( {{l_{\rm{D}}}, u} \right) = \dfrac{u}{{{\eta _{\rm{D}}}}}{\overline \psi _{{\rm{flD}}}}$ (4)

$l_{\rm{D}}$—裂缝长度，无因次；

$R_{\rm{fD}}$—裂缝导流能力，无因次；

$\eta_{\rm{D}}$—导压系数，无因次。

fl—有限导流裂缝。

 ${A_N}{\overline \psi _{{\rm{WD}}\left( N \right)}} + {B_N}{\overline \psi _{{\rm{flD}}\left( N \right)}} + {C_N}{\overline \psi _{{\rm{flD}}\left( {N + 1} \right)}} + {D_N}{\overline \psi _{{\rm{flD}}\left( {N - 1} \right)}} - {E_N}{\overline q _{{\rm{flD}}\left( N \right)}} = 0$ (5)

$A_N$$B_N$$C_N$$D_N$$E_N$—单元$N$的方程系数，与单元位置有关；

$N$—离散段编号；

${\overline \psi _{{\rm{wD}}}}$— Laplace空间下的井底拟压力，无因次。

 图2 主缝次缝交汇单元示意图 Fig. 2 Schematic diagram of intersection unit

(1) 内部裂缝单元

 $\left( { - \dfrac{{{R_{{\rm{fD}}(N)}}}}{{{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}} - \dfrac{{{R_{{\rm{fD}}(N)}}u}}{{2{\rm{ \mathsf{ π} }}{\eta _{\rm{D}}}}}} \right){\overline \psi _{{\rm{flD}}\left( N \right)}} \\ + \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}}{\overline \psi _{{\rm{flD}}\left( {N + 1} \right)}} + \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}}{\overline \psi _{{\rm{flD}}\left( {N - 1} \right)}} - {\overline q _{{\rm{flD}}\left( N \right)}} = 0$ (6)

(2) 与井筒相邻单元(如图 2中离散段1)

 $\left( { - \dfrac{{3{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}} - \dfrac{{{R_{{\rm{fD}}(N)}}u}}{{2{\rm{ \mathsf{ π} }}{\eta _{\rm{D}}}}}} \right){\overline \psi _{{\rm{flD}}\left( N \right)}} + \\ \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}}{\overline \psi _{{\rm{flD}}\left( {N + 1} \right)}} + \dfrac{{{R_{{\rm{fD}}(N)}}}}{{{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}}{\overline \psi _{{\rm{wD}}}} - {\overline q _{{\rm{flD(}}N)}} = 0$ (7)

(3) 封闭端裂缝单元(如图 2中离散段4、5、8)

 $\left( { - \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}} - \dfrac{{{R_{{\rm{fD}}(N)}}u}}{{2{\rm{ \mathsf{ π} }}{\eta _{\rm{D}}}}}} \right){\overline \psi _{{\rm{flD}}\left( N \right)}} + \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}\Delta l_{\rm{D}}^2}}{\overline \psi _{{\rm{flD}}\left( {N - 1} \right)}} - {\overline q _{{\rm{flD}}\left( N \right)}} = 0$ (8)

(4) 交汇点附近单元(如图 2中离散段2、3、6、7)

 $\dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}}}\sum\limits_{i = 1}^N {\dfrac{{{T_{{\rm{D}}i, 0}}{T_{{\rm{D}}N, 0}}}}{{\sum\limits_{k = 1}^N {{T_{{\rm{D}}k, 0}}} }}} {\overline \psi _{{\rm{flD}}(i)}}\!-\! \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}}} \\ \left( {\sum\limits_{i = 1}^N {\dfrac{{{T_{{\rm{D}}i, 0}}{T_{{\rm{D}}N, 0}}}}{{\sum\limits_{k = 1}^N {{T_{{\rm{D}}k, 0}}} }}}\!+\!\dfrac{u}{{{\eta _{\rm{D}}}}}\!+\!{T_{{\rm{D}}N + 1, N}}} \right) \\ {\overline \psi _{{\rm{flD}}(N)}}\!+\! \dfrac{{{R_{{\rm{fD}}(N)}}}}{{2{\rm{ \mathsf{ π} }}}}{T_{{\rm{D}}N\!+\! 1, N}}{\overline \psi _{{\rm{flD}}(N + 1)}}\!-\!{\overline q _{{\rm{flD}}\left( N \right)}}\!=\!0$ (9)

 ${T_{{\rm{D}}N + 1, N}} = \dfrac{1}{{\Delta {l_{{\rm{D}}(N)}}\left( {\dfrac{{{l_{{\rm{D}}(N + 1)}}}}{2} + \dfrac{{{l_{{\rm{D}}(N)}}}}{2}} \right)}} = \dfrac{1}{2}{T_{{\rm{D}}N, 0}}$ (10)

2.3 模型求解

 $\sum\limits_{i = 1}^{{N_{\rm{w}}}} { - {{\overline \psi }_{{\rm{flD}}\left( i \right)}} + {N_{\rm{w}}}} {\overline \psi _{{\rm{wD}}}} = \dfrac{{{\rm{ \mathsf{ π} }}\Delta {l_{\rm{D}}}}}{{2u{R_{{\rm{fD}}}}}}$ (11)

$N_{\rm{w}}$—与井筒相连的离散单元的个数，无因次。

3 典型曲线及压力动态特征分析

 图3 压裂水平井有限导流模型流动阶段划分图 Fig. 3 Flow stage diagram of finite conductivity model for fracturing horizontal wells

Ⅰ井储阶段：该阶段在形态上与常规曲线一致，表现为重合的斜率为1的两条曲线。Ⅱ井储后的过渡流阶段：表现为与常规曲线类似的“拱形”。Ⅲ双线性流动阶段：该阶段拟压力导数曲线表现为斜率为“1/4”的直线。Ⅳ线性流阶段：双对数曲线表现为两条斜率为1/2的平行线，此时各裂缝段间互不干扰。Ⅴ裂缝干扰流动阶段：各压裂段内，主缝与次生相互干扰。Ⅵ早期第一径向流阶段：各压裂段间互不干扰，段内主缝与其相沟通的次缝形成一个整体进行协调生产。表现出拟压力导数曲线值为1/2$m$的特点($m$—压裂段数)。Ⅶ椭圆流阶段：该阶段相邻两压裂段之间开始干扰。Ⅷ天然裂缝系统的拟径向流阶段：改造区域裂缝对曲线的影响已经结束。表现出拟压力导数曲线值为1/2的特点。Ⅸ基质向天然裂缝的窜流段：基岩中的气体解吸脱附并以拟稳态的形式向天然裂缝进行扩散补给，在双对数曲线上表现出双重介质的“凹子”特征。Ⅹ晚期总系统的拟径向流段：在双对数曲线上表现出拟压力导数曲线值为1/2的水平直线特征。

3.1 吸附解吸常数$\alpha$

 图4 吸附解吸常数$\alpha$对双对数曲线形态的影响 Fig. 4 Effect of adsorption index $\alpha$ on type curves
3.2 主缝与次缝夹角$\theta$

 图5 主缝与次缝夹角$\theta$对双对数曲线形态的影响 Fig. 5 Effect of angle $\theta$ on type curves
3.3 次生裂缝组数$m_{\rm{f}}$

 图6 次生裂缝组数$m_{\rm{f}}$对双对数曲线形态的影响 Fig. 6 Effect of the number of induced fracture groups $m_{\rm{f}}$ on type curves
3.4 裂缝导流能力$R_{\rm{fD}}$

 图7 裂缝导流能力$R_{\rm{fD}}$对双对数曲线的影响 Fig. 7 Effect of fracture conductivity $R_{\rm{fD}}$ on type curves

3.5 无因次导压系数$\eta_{\rm{fD }}$

$\eta_{\rm{fD}}$对双对数曲线形态的影响如图 8所示。从图 8中可以看出，当$\eta_{\rm{fD}}$很小时，拟压力导数曲线在过渡段之后出现了一个“凹子”，且$\eta_{\rm{fD}}$越小“凹子”越深，$\eta_{\rm{fD}}$反映出主裂缝和次生裂缝中流体的压缩性与流动能力的大小，当$\eta_{\rm{fD}}$较大时，即裂缝内流体的压缩性可忽略不计，主裂缝与次生裂缝渗透率远大于天然裂缝渗透率时，“凹子”消失。由于页岩气系统中的主裂缝与次生裂缝体积相比于整个页岩气藏体积很小，故压裂裂缝中由于气体弹性膨胀所引起的流量可忽略不计，此时$\eta_{\rm{fD}}$无穷大。

 图8 无因次导压系数$\eta_{\rm{fD}}$对双对数曲线形态的影响 Fig. 8 Effect of coefficient of pressure conductivity $\eta_{\rm{fD}}$ on type curves
4 结论

(1) 建立了考虑次生裂缝影响的有限导流裂缝流动模型，利用半解析的方法，有效解决了有限导流裂缝交点的流量分配问题。

(2) 计算得到了井底压力的解，并绘制出典型曲线，共可分为10个流动阶段。

(3) 通过引入吸附解吸常数$a$来表征页岩气吸附解吸能力，$a$越大，阶段Ⅸ的“凹子”越深。

(4) 引入了表征次生裂缝的相关参数：次生裂缝组数$m_{\rm{f}}$越多，角度$\theta$越接近90°，拟压力导数曲线越低。裂缝导流能力$R_{\rm{fD}}$越大，过渡段出现越早，双线性流阶段持续时间越短，而线性流越早出现。

 [1] JAVADPOUR F. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone)[J]. Journal of Canadian Petroleum Technology, 2009, 48(8): 16-21. doi: 10.2118/09-08-16-DA [2] NELSON P H. Pore-throat sizes in sandstones, tight sandstones, and shales[J]. AAPG Bulletin, 2009, 93(3): 329-340. doi: 10.1306/10240808059 [3] 王祥, 刘玉华, 张敏, 等. 页岩气形成条件及成藏影响因素研究[J]. 天然气地球科学, 2010, 21(2): 350-356. WANG Xiang, LIU Yuhua, ZHANG Min, et al. Conditions of formation and accumulation for shale gas[J]. Natural Gas Geoscience, 2010, 21(2): 350-356. [4] 邹才能, 董大忠, 王社教, 等. 中国页岩气形成机理、地质特征及资源潜力[J]. 石油勘探与开发, 2010, 37(6): 641-653. ZOU Caineng, DONG Dazhong, WANG Shejiao, et al. Geological characteristics, formation mechanism and resource potential of shale gas in China[J]. Petroleum Exploration and Development, 2010, 37(6): 641-653. [5] 郭晶晶.基于多重运移机制的页岩气渗流机理及试井分析理论研究[D].成都: 西南石油大学, 2013. GUO Jingjing. Research on multiple migration mechanisms and well testing theory for shale gas reservoirs[D]. Chengdu: Southwest Petroleum University, 2013. http://cdmd.cnki.com.cn/Article/CDMD-10615-1016045580.htm [6] 赵玉龙.基于复杂渗流机理的页岩气藏压裂井多尺度不稳定渗流理论研究[D].成都: 西南石油大学, 2015. ZHAO Yulong. Research on transient seepage theory of fractured wells with complex percolation mechanisms in multiscale shale gas reservoir[D]. Chengdu: Southwest Petroleum University, 2015. http://cdmd.cnki.com.cn/Article/CDMD-10615-1016005924.htm [7] BROWN M, OZKAN E, RAGHAVAN R, et al. Practical solutions for pressure-transient responses of fractured horizontal wells in unconventional shale reservoirs[J]. SPE Reservoir Evaluation & Engineering, 2011, 14(6): 663-676. doi: 10.2118/125043-PA [8] REN Zongxiao, WU Xiaodong, LIU Dandan, et al. Semianalytical model of the transient pressure behavior of complex fracture networks in tight oil reservoirs[J]. Journal of Natural Gas Science & Engineering, 2016, 35: 497-508. doi: 10.1016/j.jngse.2016.09.006 [9] ANDRADE P J F, CIVAN F, DEVEGOWDA D, et al. Design and examination of requirements for a rigorous shalegas reservoir simulator compared to current shale-gas simulator[C]. SPE 144401 MS, 2011. doi: 10.2118/144401-MS [10] OZKAN E, RAGHAVAN R S, APAYDIN O G. Modeling of fluid transfer from shale matrix to fracture network[C]. SPE 134830 MS, 2010. doi: 10.2118/134830-MS [11] 高杰.页岩气藏多段压裂水平井压力动态特征研究[D].成都: 西南石油大学, 2014. GAO Jie. Study on pressure dynamics of multi-stage fracturing horizontal wells in shale gas reservoir[D]. Chengdu: Southwest Petroleum University, 2014. http://cdmd.cnki.com.cn/Article/CDMD-10615-1014409750.htm [12] REN Junjie, GUO Ping. A novel semi-analytical model for finite-conductivity multiple fractured horizontal wells in shale gas reservoirs[J]. Journal of Natural Gas Science and Engineering, 2015, 24: 35-51. doi: 10.1016/j.jngse.2015.-03.015 [13] 魏明强.页岩气藏压裂水平井数值试井及产量递减分析理论研究[D].成都: 西南石油大学, 2016. WEI Mingqiang. Research on numerical well testing and production decline analysis theory for multi-fractured horizontal wells in shale gas reservoirs[D]. Chengdu: Southwest Petroleum University, 2016. [14] ZHAO Yulong, ZHANG Liehui, XIONG Ye, et al. Pressure response and production performance for multifractured horizontal wells with complex seepage mechanism in box-shaped shale gas reservoir[J]. Journal of Natural Gas Science and Engineering, 2016, 32: 66-80. doi: 10.1016/j.jngse.2016.04.037 [15] CHANG Jincai, YORTSOS Y C. Pressure transient analysis of fractal reservoirs[J]. SPE Formation Evaluation, 1990, 5(1): 31-38. doi: 10.2118/18170-PA [16] 孙致学, 姚军, 樊冬艳, 等. 基于离散裂缝模型的复杂裂缝系统水平井动态分析[J]. 中国石油大学学报(自然科学版), 2014, 38(2): 109-115. SUN Zhixue, YAO Jun, FAN Dongyan, et al. Dynamic analysis of horizontal wells with complex fractures based on a discrete-fracture model[J]. Journal of China University of Petroleum, 2014, 38(2): 109-115. doi: 10.3969/j.issn.1673-5005.2014.02.017 [17] 贾品, 程林松, 黄世军, 等. 压裂裂缝网络不稳态流动半解析模型[J]. 中国石油大学学报(自然科学版), 2015, 39(5): 107-116. JIA Pin, CHENG Linsong, HUANG Shijun, et al. A semianalytical model for transient flow behavior of hydraulic fracture networks[J]. Journal of China University of Petroleum, 2015, 39(5): 107-116. doi: 10.3969/j.issn.1673-5005.2015.05.015 [18] ZENG Fanhua, ZHAO Gang, LIU Hong. A new model for reservoirs with a discrete-fracture system[J]. Journal of Canadian Petroleum Technology, 2012, 51(2): 127-136. doi: 10.2118/150627-PA [19] JIA Pin, CHENG Linsong, HUANG Shijun, et al. Transient behavior of complex fracture networks[J]. Journal of Petroleum Science and Engineering, 2015, 132: 1-17. doi: 10.1016/j.petrol.2015.04.041 [20] 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[J]. 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[J]. Chinese Journal of Computational Physics, 2015, 32(6): 685-692. doi: 10.3969/j.issn.1001-246X.2015.06.008 [21] LANGMUIR I. The adsorption of gases on plane surfaces of glass, mica and platinum[J]. Journal of the American Chemical Society, 1918, 40(9): 1361-1403. doi: 10.1021/ja02242a004 [22] 李政澜.页岩气藏压裂水平井试井分析理论研究[D].成都: 西南石油大学, 2017. LI Zhenglan. Research on well testing theory of fractured horizontal wells for shale gas reservoirs[D]. Chengdu: Southwest Petroleum University, 2017. http://cdmd.cnki.com.cn/Article/CDMD-10615-1017108287.htm