A Green’s Function Construction Method of the Single Well Seepage Model for Asymmetric Fractures
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摘要:
不对称裂缝渗流规律可借助Green函数方法进行求解。根据基本渗流理论，建立了不对称裂缝点源数学模型，采用无因次化与Laplace变换，得到了Laplace空间的无因次点源数学模型微分方程。将未知Green函数与点源微分方程相结合，并考虑点源微分方程的齐次条件以及点源微分方程的特征，给出了如何构造Green函数使之满足点源微分方程齐次边界以及未知目标函数求解的一般方法。根据空间Green函数的对称性和连续性，得出了不对称裂缝点源模型Green函数的形式。最后通过不对称裂缝压裂直井渗流数学模型，验证了该文给出的Green函数两种形式与文献和商业试井分析软件Saphir的数值计算结果一致。
Abstract:The seepage law for asymmetric fractures can be solved by the Green’s function method. According to the basic seepage theory, the point source mathematical model for asymmetric fractures was established. The dimensionless point source mathematical model differential equation in the Laplacian space was obtained through the dimensionless transformation and the Laplacian transformation. By means of the unknown Green’s function combined with the point source differential equation, and in view of the homogeneous boundary conditions for the point source differential equation and the characteristics of the point source differential equation, a general construction method for Green’s function was given to meet the homogeneous boundary conditions for the point source differential equation and the solution of the unknown objective function. According to the symmetry and continuity of spatial Green’s function, the Green form of the asymmetric fracture point source model was obtained. Finally, through the seepage mathematical model for the asymmetric-fracture vertical well, it was verified that the 2 forms of Green’s function are consistent with the results calculated in references and with the commercial well test analysis software Saphir.
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Key words:
- Green’s function /
- asymmetric fracture /
- fractured well /
- unequal-spacing grid
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图 1 不对称裂缝点源的几何模型
Figure 1. The geometric model for the asymmetric fracture point source
图 2 对称裂缝不同形式Green函数计算结果对比
Figure 2. Comparison of calculation results from different forms of Green’s function for the symmetric fracture
图 3 不对称裂缝不同形式Green函数计算结果对比
Figure 3. Comparison of calculation results from different forms of Green’s function for the asymmetric fracture
图 5 解析解与试井分析软件Spahir数值计算结果对比
Figure 5. Comparison between the analytical solution and the numerical calculation results with well test analysis software Saphir
图 6 对称裂缝等距网格与不等距网格计算结果对比
Figure 6. Comparison of calculation results between the equal-spacing grid and the unequal spacing grid for the symmetric fracture
图 7 不对称裂缝等距网格与不等距网格计算结果对比
Figure 7. Comparison of calculation results between the equal-spacing grid and the unequal spacing grid for the asymmetric fracture
符号说明 $ {x_{\text{w}}} $ 井偏离裂缝中心位置的位移，m $ {L_{\text{F}}} $ 裂缝的半长，m $ {W_{\text{F}}} $ 裂缝的宽度，m $ {k_{\text{F}}} $ 裂缝的渗透率，m2 $ {p_{\text{F}}} $ 裂缝中流体压力，Pa $ \mu $ 流体的黏度，Pa.s $ q $ 1/2裂缝单元总流量，m3/s $ h $ 储层厚度，m $ {L_{{\text{ref}}}} $ 参考长度，一般取裂缝的半长，m ${\text{δ} }\left( {x - {x_{\text{w} } } } \right)$ Dirac函数 $ k $ 储层的有效渗透率，m2 $ G\left( {{x_{\text{D}}};\alpha } \right) $ 空间位置Green函数 $ s $ Laplace变量 $ {G'}\left( {{x_{\text{D}}};\alpha } \right) $ Green函数一阶导数 $ p $ 地层中任意位置的压力，Pa ${G'' }\left( { {x_{\text{D} } };\alpha } \right)$ Green函数二阶导数 $ {x_{{\text{asmy}}}} $ 不对称因子，无因次 $ \eta \left( {{x_{\text{D}}} - \alpha } \right) $ 单位阶跃函数 $ \alpha $ 空间位置Green函数原像位置，m $ {C_{{\text{FD}}}} $ 无因次裂缝导流能力，无因次 $ {q_{\text{F}}} $ 单位长度裂缝线流量，m3/s $ {{\text{K}}_0}\left( x \right) $ 第二类零阶修正Bessel函数 ${p_{\rm{i}}}$ 原始地层压力，Pa $ {{\text{I}}_0}\left( x \right) $ 第一类零阶Bessel函数 $ {r_{{\text{eD}}}} $ 外边界无因次半径，无因次 $ {x_{{\text{midD}}\left( i \right)}} $ 裂缝离散第i个网格的中点 $ {x_{{\text{D}}\left( j \right)}} $ 裂缝离散第j个网格的左边界 $ {x_{{\text{D}}\left( {j + 1} \right)}} $ 裂缝离散第j个网格的右边界 $ {(\cdot)_{\text{D}}} $ 无因次变量 $ \overline{(\cdot)} $ Laplace空间参数 
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