地球科学  2017, Vol. 42 Issue (8): 1324-1332. PDF     0

1. 合肥工业大学数学学院, 安徽合肥 230009;
2. 中国石油大庆油田测试技术服务分公司, 黑龙江大庆 163453;
3. 中国科学技术大学工程科学学院, 安徽合肥 230026

Numerical Well Test Interpretation of Massive Multistage Fractured Horizontal Wells in Tight Oil Reservoirs and Effect of Permeability of Exterior Region on Well Test Curves
Li Daolun1 , Yang Jinghai2 , Yan Shu2 , Zha Wenshu1 , Lu Detang3 , Zeng Yishan1
1. School of Mathematics, Hefei University of Technology, Hefei 230009, China;
2. Logging and Testing Services Company, Daqing Oilfield Company, PetroChina, Daqing 163453, China;
3. School of Engineering Science, University of Science and Technology of China, Hefei 230026, China
Abstract: Numerical well test interpretation of massive multistage fractured horizontal wells can be used for the fracturing effect evaluation, which is very important for productivity evaluation. However, few field case studies have been conducted. Numerical solution of oil-water two-phase flow based on PEBI grid and description of stimulated reservoir volume (SRV) by main fractures with infinite conductivity and improved permeability of the region with minor fractures are used in combination to interpret the transient pressure of massive multistage fractured horizontal wells in tight oil reservoirs in Daqing oilfield. The interpretation and sensitivity analysis of permeability of exterior region show that the value of pressure derivative becomes smaller at early time and becomes larger at late time when the permeability of exterior region decreases, compared with pressure derivative without permeability modification, which indicates a turning point on the curves of pressure derivative. Prior to the turning point, the pressure derivative with smaller permeability in exterior region is relatively smaller, while after the point, the pressure derivative is larger. The time of appearance of the turning point is related to the magnitude of permeability between those of exterior and interior regions. Therefore, when good fitting of pressure derivative curves achieves at early time and bad fitting of pressure derivative curves at late time, adjustment of permeability of exterior region cannot wholly improve the fitting effect, and other parameters need to be adjusted to improve the fitting. This study can facilitate future transient pressure analysis for massive multistage fractured horizontal wells in tight oil reservoirs.
Key Words: multistage fractured horizontal well    stimulated reservoir volume    fracture half length    perpendicular bisector grid    numerical solution    oil-water two-phase    petroleum geology

0 引言

1 数学模型

 $\nabla \cdot \left[ {\frac{{K{K_{{\rm{ro}}}}}}{{{\mu _{\rm{o}}}{B_{\rm{o}}}}}\left({\nabla {p_{\rm{o}}} - {\gamma _{\rm{o}}}\nabla Z} \right)} \right] = \frac{\partial }{{\partial t}}\left({\frac{{\varphi {S_{\rm{o}}}}}{{{B_{\rm{o}}}}}} \right) - {q_{{\rm{osc}}}},$ (1)

 $\nabla \cdot \left[ {\frac{{K{K_{{\rm{rw}}}}}}{{{\mu _{\rm{w}}}{B_{\rm{w}}}}}\left({\nabla {p_{\rm{w}}} - {\gamma _{\rm{w}}}\nabla Z} \right)} \right] = \frac{\partial }{{\partial t}}\left({\frac{{\varphi {S_{\rm{w}}}}}{{{B_{\rm{w}}}}}} \right) - {q_{{\rm{wsc}}}},$ (2)

 ${q_{{\rm{osc}}}} = \frac{{2{\rm{\pi }}K{K_{{\rm{ro}}}}h}}{{{\mu _{\rm{o}}}{B_{\rm{o}}}\left[ {\ln \left({{r_{\rm{e}}}/{r_{\rm{w}}}} \right) + S} \right]}}\left({{p_i} - {p_{{\rm{wf}}}}} \right),$ (3)

 ${q_{{\rm{wsc}}}} = \frac{{2{\rm{\pi }}K{K_{{\rm{rw}}}}h}}{{{\mu _{\rm{w}}}{B_{\rm{w}}}\left[ {\ln \left({{r_{\rm{e}}}/{r_{\rm{w}}}} \right) + S} \right]}}\left({{p_i} - {p_{{\rm{wf}}}}} \right),$ (4)

 $\begin{array}{l} \sum\limits_{l = {\rm{o, w}}} {\frac{{2{\rm{\pi }}kh}}{{{\mu _l}{B_l}\left[ {\ln \left({{r_{\rm{e}}}/{r_{\rm{w}}}} \right) + S} \right]}}\left({{p_i} - {p_{{\rm{wf}}}}} \right)} - \\ \frac{C}{{\Delta t}}\left({p_{{\rm{wf}}}^{n + 1} - p_{{\rm{wf}}}^n} \right) = Q, \end{array}$ (5)

PEBI网格的对偶网格是Delaunary三角网格，相邻网中心点的连线与邻边垂直平分(李道伦和查文舒，2013).多段压裂水平井的非结构PEBI网格划分可参见Li et al.(2014).这里以油相为例，给出数值计算格式推导过程.设网格i为当前网格，采用控制体积法，对强非线性项采用线性隐式化，对弱非线性项采用显式线性化，采用上游加权方式，则有：

 $\begin{array}{l} \sum\limits_j {T_{ij, {\rm{o}}}^n} \left({{\rm{\delta }}{p_j} - {\rm{\delta }}{p_i} + p_j^n - p_i^n} \right) + \sum\limits_j {\left({\frac{{\partial {T_{ij, {\rm{o}}}}}}{{\partial {S_{\rm{w}}}}}} \right)} _ + ^n \cdot \\ \;\;\left({p_j^n - p_i^n} \right){\rm{\delta }}{S_{{\rm{w, + }}}} = \left({{C_{{\rm{op}}}}{\rm{\delta }}{p_i} + {C_{{\rm{ow}}}}{\rm{\delta }}{S_{{\rm{w, }}i}}} \right) + q_{{\rm{osc}}}^{n + 1}, \end{array}$ (6)

 $\begin{array}{l} \;\;\;{q_{\rm{o}}} = \\ \frac{{W{I_{{\rm{o, }}i}}\lambda _{{\rm{o, }}i}^{n + 1}}}{{\sum\limits_j {W{I_j}\left({\lambda _{{\rm{o, }}j}^{n + 1} + \lambda _{{\rm{w, }}j}^{n + 1}} \right) + \frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}} }}\left[ {\frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}p_i^{n + 1} - \frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}p_{{\rm{wf}}}^n + Q} \right], \end{array}$ (7)

 $q_{\rm{o}}^{n + 1} = q_{\rm{o}}^n + \frac{{\partial q_{\rm{o}}^n}}{{\partial p}}{\rm{\delta }}{p_i} + \frac{{\partial q_{\rm{o}}^n}}{{\partial {S_{{\rm{w, }}i}}}}{\rm{\delta }}{S_{{\rm{w, }}i}},$ (8)

 $\begin{array}{l} \sum\limits_j {T_{ij, {\rm{o}}}^n\left({{\rm{\delta }}{p_j} - {\rm{\delta }}{p_i} + p_j^n - p_i^n} \right) + \sum\limits_j {\left({\frac{{\partial {T_{ij, {\rm{o}}}}}}{{\partial {S_{\rm{w}}}}}} \right)} _ + ^n \cdot } \\ \;\left({p_j^n - p_i^n} \right){\rm{\delta }}{S_{{\rm{w, + }}}} = \left({{C_{{\rm{op}}}}{\rm{\delta }}{p_i} + {C_{{\rm{ow}}}}{\rm{\delta }}{S_{{\rm{w, }}i}}} \right) + \\ {C_{{\rm{wop}}}}{\rm{\delta }}{p_i} + {C_{{\rm{owp}}}}\left[ {\left({p_i^n - p_{{\rm{wf}}}^n} \right) - Q\frac{{\Delta t}}{{C \cdot {f_{\rm{o}}}}}} \right] + \frac{{\partial {q_{\rm{o}}}}}{{\partial {S_{\rm{w}}}}}\partial {S_{{\rm{w, }}i}}, \end{array}$ (9)

 $\begin{array}{*{20}{l}} {{C_{{\rm{wop}}}} = \frac{{W{I_{{\rm{o}},i}}\lambda _{{\rm{o}},i}^n}}{{\sum\limits_j {W{I_j}\left( {\lambda _{{\rm{o}},j}^n + \lambda _{{\rm{w}},j}^n} \right) + \frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}} }}\frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}};}\\ {\;\;\frac{{\partial {q_{\rm{o}}}}}{{\partial {S_{\rm{w}}}}} = }\\ {\frac{{W{I_i}\lambda _{{\rm{o}},i}^{n'}\left( {\sum\limits_j {W{I_j}\left( {\lambda _{{\rm{o}},j}^n + \lambda _{{\rm{w}},j}^n} \right) + \frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}} } \right) - \sum\limits_j {W{I_j}\left( {\lambda _{{\rm{o}},j}^{n'} + \lambda _{{\rm{w}},j}^{n'}} \right) \cdot W{I_i}\lambda _{{\rm{o}},i}^n} }}{{{{\left( {\sum\limits_j {W{I_j}\left( {\lambda _{{\rm{o}},j}^n + \lambda _{{\rm{w}},j}^n} \right) + \frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}} } \right)}^2}}}.} \end{array}$

$\left[ {\frac{{C \cdot {f_{\rm{o}}}}}{{\Delta t}}\left({p_i^{n + 1} - p_{{\rm{wf}}}^n} \right) + Q} \right]$方程(9) 中，若网格i与井不相邻，则所有与井相关的项都为0，即Cwop$\frac{{\partial {q_{\rm{o}}}}}{{\partial {S_{\rm{w}}}}}$皆为0.

2 压裂改造区域描述方法

 Download: larger image 图 1 等效主裂及渗透率提高的区域示意 Fig. 1 Schematic of equivalent main fracture and permeability improved region

 Download: larger image 图 2 PEBI网格划分与主裂缝 Fig. 2 PEBI gridding and equivalent main fracture
3 正确性验证

 Download: larger image 图 3 相渗曲线 Fig. 3 Curves of relative permeability

 Download: larger image 图 4 所解释的油藏示意 Fig. 4 The sketch of the interpreted reservoir a.油藏面积约为4 640 m×3 900 m；b.油藏厚度为1.7 m
 Download: larger image 图 5 压力拟合结果 Fig. 5 Pressure fitting result a.压力变化及其导数拟合情况；b.压力史拟合情况

4 复合区域渗透率对曲线的影响

 Download: larger image 图 6 不同K1与K2组合下的瞬态压力响应特征 Fig. 6 Characteristics of pressure transient response under different combination of K1 and K2 a.全局图；b.压力导数曲线局部放大

 Download: larger image 图 7 外区渗透率K2=4.8 mD与K2=0.8 mD的瞬态压力曲线对比 Fig. 7 Comparison of pressure transient curves between permeability of exterior region K2=4.8 mD and K2=0.8 mD a.全局图；b.对方形框的局部放大

 Download: larger image 图 8 外区渗透率K2=0.8 mD与K2=0.1 mD的瞬态压力曲线对比 Fig. 8 Comparison of pressure transient curves between permeability of exterior region K2=4.8 mD and K2=0.1 mD a.全局图；b.对方形框的局部放大

 Download: larger image 图 9 外区渗透率K2=0.01 mD与K2=0.1 mD的瞬态压力曲线对比 Fig. 9 Comparison of pressure transient curves between permeability of exterior region K2=0.01 mD and K2=0.1 mD a.外区渗透率K2对瞬态压力响应的影响；b.局部放大
 Download: larger image 图 10 可通过调整外区渗透率来拟合的情形 Fig. 10 Enable to fit by adjusting permeability of exterior region