Abstract
As an important stimulation technology, refracturing is commonly used to enhance oil or gas recovery in tight reservoirs, particularly for poorly/damaged fractured wells in these reservoirs. However, the stimulation effect of refracturing is not clear at present. Accurate evaluation of the effect of refracturing treatment on the productivity of a poorly/damaged fractured well is of great significance to the rapid and efficient development of tight reservoirs. In this study, two semi-analytical models, including an initially fractured well model and a refractured well model, were established to calculate their productivity indices, respectively, for quantitative productivity evaluation. Productivity index ratio, a newly defined parameter reflecting the characteristics of both productivity indices, was calculated to evaluate the refracturing stimulation effect. The comparisons with two classic cases indicated our solutions were exactly consistent with the results in the previous literature and verified the reliability and accuracy of our solutions. Results show that compared with initially fractured well, the productivity of refractured well first rises rapidly and then slows down as principal/reoriented fracture angle or rotation angle increases, but an inverse relationship may occur at low initial fracture conductivity. Meanwhile, the productivity index ratio curves under the same initial fracture conductivity converge into a bunch of curves, first increasing and then slowing down as reorientation fracture conductivity enhances. Overall, refracturing treatment is more effective for fractured wells when initial fracture conductivity becomes lower. Reoriented/principal fracture length ratio has a weak negative correlation with the productivity index ratio, but reorientation fracture number shows a stronger influence on the stimulation than the other factors. Principal/reoriented fracture angle, fracture length ratio, reorientation fracture number and fracture conductivity should be optimized before refracturing to achieve the optimal productivity. Additionally, in a rectangular reservoir, the central refractured well obtains larger productivity than the eccentric refractured well, and hydraulic fractures deploying along the reservoir extension can slightly increase well productivity.
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Abbreviations
- x :
-
Distance in the x-axis, m
- y :
-
Distance in the y-axis, m
- x e :
-
Reservoir length in the x-axis, m
- y e :
-
Reservoir length in the y-axis, m
- h :
-
Reservoir thickness, m
- L f 1 :
-
Principal fracture length, m
- L f 2, L f 3 :
-
Reoriented fracture length, m
- L R :
-
Reference length, m
- l :
-
Discrete segment length, m
- w f :
-
Fracture width, m
- k :
-
Permeability in the reservoir, 10−3μm2
- k f 1 :
-
Permeability in the initial fracture, 10−3μm2
- k f 2 :
-
Permeability in the reorientation fracture, 10−3μm2
- θ p :
-
Principal fracture angle, degree
- θ r 1, θ r 2 :
-
Reoriented fracture angle, degree
- p i :
-
Initial reservoir pressure, MPa
- p avg :
-
Average reservoir pressure, MPa
- p w :
-
Wellbore pressure, MPa
- φ :
-
Reservoir porosity, fraction
- μ :
-
Fluid viscosity, mPa s
- c t :
-
Total compressibility, 1/MPa
- β :
-
Reoriented/principal fracture length ratio, fraction
- Q :
-
Wellbore flow rate, m3/d
- q fw :
-
Flow rate of per unit fracture length, m3/d
- q f :
-
Flow rate strength, m2/d
- x D :
-
Dimensionless distance in the x-axis
- y D :
-
Dimensionless distance in the y-axis
- x eD :
-
Dimensionless reservoir length in the x-axis
- y eD :
-
Dimensionless reservoir width in the y-axis
- x′ D :
-
Dimensionless coordinate transformation of xD along reorientation fracture
- y′ D :
-
Dimensionless coordinate transformation of yD along reorientation fracture
- p D :
-
Dimensionless reservoir pressure
- p 1 D :
-
Dimensionless pressure in the initially fractured well model
- p 2 D :
-
Dimensionless pressure in the refractured well model
- p 1 D i :
-
Dimensionless pressure of the ith minute segment caused by the production of the initially fractured well
- p 2 D i :
-
Dimensionless pressure of the ith minute segment caused by the production of the refractured fractured well
- p avgD :
-
Dimensionless average reservoir pressure
- p wD :
-
Dimensionless wellbore pressure
- q D :
-
Dimensionless point source flux
- q PfD :
-
Dimensionless flow rate strength of the initial fracture
- q RfD :
-
Dimensionless flow rate strength of the reorientation fracture
- q PDk :
-
Dimensionless flow flux of the kth minute fracture segment in the initial fracture
- q RDm :
-
Dimensionless flow flux of the mth minute fracture segment in the reorientation fracture
- J PD :
-
Dimensionless productivity index for the initially fractured well
- J RD :
-
Dimensionless productivity index for the refractured well
- F D :
-
Pseudo-steady-state pressure distribution function
- F DP :
-
Pseudo-steady-state pressure distribution function for the initially fractured well
- F DR :
-
Pseudo-steady-state pressure distribution function for the refractured well
- R JD :
-
Productivity index ratio
- C fD 1 :
-
Dimensionless initial fracture conductivity
- C fD 2 :
-
Dimensionless reorientation fracture conductivity
- l D :
-
Dimensionless discrete segment length
- t D :
-
Dimensionless time
- χ D :
-
Dimensionless integral variable
- N PF :
-
Total discrete segments of the initial fracture
- N RF :
-
Total discrete segments of reorientation fracture
- △l D i :
-
Dimensionless length of the ith discrete segment, i = 1, 2, 3…
- sinh:
-
Hyperbolic sine function
- cosh:
-
Hyperbolic cosine function
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Acknowledgements
This study was financially supported by the National Natural Science Foundation of China (Program No. 52004038), the CNPC Innovation Foundation (Program No. 2020D-5007-0201), the Natural Science Basic Research Program of Shaanxi (Program No. 2021JQ-596) and the National Science and Technology Major Project of China (Program No. 2016ZX05046-003).
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Contributions
GX proposed the idea, established the mathematical model and wrote the main manuscript. MW made substantial contributions to the modeling and conceptualization of the work. YZ, HS, WG, TL, RL and XD contributed to the programming, result analysis and discussion.
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The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Appendix: Fast Algorithm of Eq. 11
Appendix: Fast Algorithm of Eq. 11
In order to get the rapid speed of computation, Eq. 11 needs further treatment to obtain its fast algorithm.
-
(1)
When \(y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} \ge 0\), Eq. 11 can be rewritten as:
$$\begin{aligned} & F_{{\mathrm{D}}} \left( {x_{{\mathrm{D}}} ,y_{{\mathrm{D}}} ,x_{{{\mathrm{D}}i}} ,y_{{{\mathrm{D}}i}} } \right) = \frac{\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {\frac{2}{3}y_{{{\mathrm{eD}}}} - 2y_{{{\mathrm{D}}i}} + \frac{{\left( {y_{{\mathrm{D}}}^{2} + y_{{{\mathrm{D}}i}}^{2} } \right)}}{{y_{{{\mathrm{eD}}}} }} + \frac{{\Delta^{2} l_{{{\mathrm{D}}i}} \sin^{2} \theta_{i} }}{{12y_{{{\mathrm{eD}}}} }}} \right] + \frac{{2x_{{{\mathrm{eD}}}} }}{{\pi \Delta l_{{{\mathrm{D}}i}} }}\sum\limits_{n = 1}^{\infty } {\frac{{\cos \left( {\frac{{n\pi x_{{\mathrm{D}}} }}{{x_{{{\mathrm{eD}}}} }}} \right)}}{{n^{2} \sinh \frac{{n\pi y_{{{\mathrm{eD}}}} }}{{x_{{{\mathrm{eD}}}} }}}}} \\ & \quad \cdot \left[ \begin{gathered} \cos \theta_{i} \left[ {\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]{ + }\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\sin \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{{\mathrm{x}}_{{{\mathrm{eD}}}} }} \hfill \\ - \cos \theta_{i} \left[ {\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]{ + }\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\sin \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ {\mathrm{ + sin}}\theta_{i} \left[ {\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]{ + }\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\cos \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ - {\mathrm{sin}}\theta_{i} \left[ {\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]{ + }\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\cos \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ \end{gathered} \right] \\ \end{aligned}$$(A-1) -
(2)
When \(y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} \le 0\), Eq. 11 can be rewritten as:
$$\begin{aligned} & F_{{\mathrm{D}}} \left( {x_{{\mathrm{D}}} ,y_{{\mathrm{D}}} ,x_{{{\mathrm{D}}i}} ,y_{{{\mathrm{D}}i}} } \right) = \frac{\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {\frac{{2y_{{{\mathrm{eD}}}} }}{3} - 2y_{{\mathrm{D}}} + \frac{{\left( {y_{{\mathrm{D}}}^{2} + y_{{{\mathrm{D}}i}}^{2} } \right)}}{{y_{{{\mathrm{eD}}}} }} + \frac{{\Delta l_{{{\mathrm{D}}i}}^{2} \sin^{2} \theta_{i} }}{{12y_{{{\mathrm{eD}}}} }}} \right] + \frac{{2x_{{{\mathrm{eD}}}} }}{{\Delta l_{{{\mathrm{D}}i}} \pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\cos \left( {\frac{{n\pi x_{{\mathrm{D}}} }}{{x_{{{\mathrm{eD}}}} }}} \right)}}{{n^{2} \sinh \frac{{n\pi y_{{{\mathrm{eD}}}} }}{{x_{{{\mathrm{eD}}}} }}}}} \\ & \quad \cdot \left[ \begin{gathered} \cos \theta_{i} \left[ {\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] + \cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\sin \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ - \cos \theta_{i} \left[ {\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] + \cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\sin \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ {\mathrm{ + sin}}\theta_{i} \left[ {\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] - \sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\cos \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ + {\mathrm{sin}}\theta_{i} \left[ {\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] - \sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\cos \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ \end{gathered} \right] \\ \end{aligned}$$(A-2) -
(3)
When \(y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} < 0 < y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i}\), Eq. 11 can be rewritten as:
$$\begin{aligned} & F_{{\mathrm{D}}} \left( {x_{{\mathrm{D}}} ,y_{{\mathrm{D}}} ,x_{{{\mathrm{D}}i}} ,y_{{{\mathrm{D}}i}} } \right) = \frac{\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {\frac{{2y_{{{\mathrm{eD}}}} }}{3} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} } \right) + \frac{{\left( {y_{{\mathrm{D}}}^{2} + y_{{{\mathrm{D}}i}}^{2} } \right)}}{{y_{{{\mathrm{eD}}}} }} + \frac{{\Delta l_{{{\mathrm{D}}i}}^{2} \sin^{2} \theta_{i} }}{{12y_{{{\mathrm{eD}}}} }} - \frac{{\left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} } \right)^{2} }}{{\Delta l_{{{\mathrm{D}}i}} \sin \theta_{i} }} - \frac{{\Delta l_{{{\mathrm{D}}i}} \sin \theta_{i} }}{4}} \right] + \frac{{2x_{{{\mathrm{eD}}}} }}{{\Delta l_{{{\mathrm{D}}i}} \pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\cos \left( {\frac{{n\pi x_{{\mathrm{D}}} }}{{x_{{{\mathrm{eD}}}} }}} \right)}}{{n^{2} \sinh \frac{{n\pi y_{{{\mathrm{eD}}}} }}{{x_{{{\mathrm{eD}}}} }}}}} \\ & \quad \cdot \left[ \begin{gathered} \cos \theta_{i} \left[ {\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] + \cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\sin \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{eD} }} \hfill \\ - \cos \theta_{i} \left[ {\cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] + \cosh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\sin \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{eD} }} \hfill \\ - \sin \theta_{i} \left[ {\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{{\mathrm{D}}i}} - y_{{\mathrm{D}}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] + \sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\cos \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{x_{{{\mathrm{eD}}}} }} \hfill \\ - \sin \theta_{i} \left[ {\sinh \frac{n\pi }{{x_{{{\mathrm{eD}}}} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} + \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right] - \sinh \frac{n\pi }{{x_{eD} }}\left[ {y_{{{\mathrm{eD}}}} - \left( {y_{{\mathrm{D}}} + y_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\sin \theta_{i} } \right)} \right]} \right]\cos \frac{{n\pi \left( {x_{{{\mathrm{D}}i}} - \frac{{\Delta l_{{{\mathrm{D}}i}} }}{2}\cos \theta_{i} } \right)}}{{{\mathrm{x}}_{{{\mathrm{eD}}}} }} \hfill \\ \end{gathered} \right] + \frac{{2\pi x_{{{\mathrm{eD}}}} \sin \theta_{i} }}{{\Delta l_{{{\mathrm{D}}i}} }} \\ & \quad \cdot \left[ {\frac{1}{3} - \frac{{\left[ {\left| {x_{{{\mathrm{D}}i}} + x_{{\mathrm{D}}} + \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} } \right)\cot \theta_{i} } \right| + \left| {x_{{{\mathrm{D}}i}} - x_{{\mathrm{D}}} + \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} } \right)\cot \theta_{i} } \right|} \right]}}{{2x_{{{\mathrm{eD}}}} }} + \frac{{\left[ {\left( {x_{{{\mathrm{D}}i}} + x_{{\mathrm{D}}} + \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} } \right)\cot \theta_{i} } \right)} \right]^{2} + \left[ {\left( {x_{{{\mathrm{D}}i}} - x_{{\mathrm{D}}} + \left( {y_{{\mathrm{D}}} - y_{{{\mathrm{D}}i}} } \right)\cot \theta_{i} } \right)} \right]^{2} }}{{4x_{{{\mathrm{eD}}}}^{2} }}} \right] \\ \end{aligned}$$(A-3)
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Xing, G., Wang, M., Zhang, Y. et al. Productivity Evaluation of Refracturing to a Poorly/Damaged Fractured Well in a Tight Reservoir. Arab J Sci Eng 47, 11361–11384 (2022). https://doi.org/10.1007/s13369-021-06056-x
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DOI: https://doi.org/10.1007/s13369-021-06056-x