*Article* **Proppant Transportation in Cross Fractures: Some Findings and Suggestions for Field Engineering**

#### **Yan Zhang 1,2, Xiaobing Lu 1,2, Xuhui Zhang 1,2,\* and Peng Li 1,2**


Received: 10 August 2020; Accepted: 18 September 2020; Published: 19 September 2020

**Abstract:** The proppant transportation is a typical two-phase flow process in a complex cross fracture network during hydraulic fracturing. In this paper, the proppant transportation in cross fractures is investigated by the computational fluid dynamics (CFD) method. The Euler–Euler two-phase flow model and the kinetic theory of granular flow (KTGF) are adopted. The dimensionless controlling parameters are derived by dimensional analysis. The equilibrium proppant height (EPH) and the ratio of the proppant mass (RPM) in the secondary fracture to that in the whole cross fracture network are used to describe the movement and settlement of proppants in the cross fractures. The main features of the proppant transportation in the cross fractures are given, and several relative suggestions are presented for engineering application in the field. The main controlling dimensionless parameters for relative EPH are the proppant Reynolds number and the inlet proppant volume fraction. The dominating dimensionless parameters for RPM are the relative width of the primary and the secondary fracture. Transportation of the proppants with a certain particle size grading into the cross fractures may be a good way for supporting the hydraulic fractures.

**Keywords:** proppant transportation; cross fractures; CFD simulation; dimensional analysis; equilibrium proppant height
