**4. Conclusions**

In this paper, the proppant transportation in the cross fractures is investigated by using the computational fluid dynamics (CFD) method. The Euler–Euler two-phase flow model and the KTFG approach are adopted to describe the flow behaviors. The dimensionless parameters relating to the proppant transportation in the cross fractures, such as the inlet proppant volume fraction, the proppant Reynolds number, the Archimedes number, the bypass angle, the relative width of the primary and secondary fracture, and the relative distance of the secondary fracture to the primary fracture entrance, are derived based on dimensional analysis.

Two dimensionless parameters are proposed to evaluate the distribution of proppants in the cross fractures, i.e., the relative EPH and the RPM. The simulation results show that the main controlling dimensionless parameters for the relative EPH are the inlet proppant volume fraction and proppant Reynolds number. The dominating dimensionless parameters for the RPM are the relative width of the primary and the secondary fracture. The relative EPH decreases with the increase of *Re*, while it increases with the increase of the sand ratio. The admirable sand ratio is about 8% for the field engineering based on the simulation results. When *wa*/*ds* changes from 6 to 11, the RPM decreases from 16% to 3%. The RPM increases from 3% to 9% when *wb*/*ds* changes from 2.4 to 5. It is suggested that the proppants with a certain particle size grading may be a good way for improving the hydraulic fracturing and increasing the oil and gas recovery. The settling effect is dominating in the secondary fracture. The proppants enter the secondary fracture mainly under the water-carrying effect. A graph (Figure 8) is given for the engineers to predict the percentage of the proppant entering the secondary fracture.

**Author Contributions:** Conceptualization, Y.Z., X.L. and X.Z.; methodology, Y.Z. P.L. and X.Z.; software, Y.Z. and P.L.; validation, Y.Z. and P.L.; investigation, Y.Z. and X.L.; data curation, Y.Z.; writing—original draft preparation, Y.Z.; writing—review and editing, X.Z. and X.L.; visualization, Y.Z. and P.L.; supervision, X.Z. and X.L.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Major Oil and Gas Projects of China, gran<sup>t</sup> number 2017X05049003-002 and The APC was funded by National Major Oil and Gas Projects of China.

**Acknowledgments:** This research is supported by the National Major Oil and Gas Projects of China (No.2017X05049003-002). The support is gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.


**Table A1.** All the cases in the numerical simulation.

**Appendix A**


