**1. Introduction**

Unconventional energy resources such as low permeability, shale, and tight oil and gas reservoirs account for a larger and larger proportion in the present oil and gas exploration [1–4]. The hydraulic fractures are the main flow channel for these fluid resources due to the natural poor flow capability of the porous media, and it is of grea<sup>t</sup> importance to know the effective support range and the distribution of proppants in cross fractures.

Many researchers have studied the proppant transportation in the cross fractures by experiments and numerical simulations. Alotaibi and Miskimins [5] designed a cross fracture system with one primary fracture, three secondary fractures, and two tertiary fractures. They found that the proppants were able to flow into the subsidiary fractures and form a proppant bed. However, they did not realize that the proppants moved not so far in the subsidiary fractures. The transportation distance of the proppant in the subsidiary fractures is important for the production. Sahai et al. [6] investigated the effects of the fracture geometrical complexity, the pumping rates, the proppant concentration, and the proppant size on the proppant transportation. The mechanism of the proppant from the primary fracture into the secondary fracture was also analyzed. McClure [7] analyzed in detail the formation process of the equilibrium proppant height (EPH). As the proppants settle at the bottom of the fractures, the height of the proppant bed gradually grows to EPH during transportation [7]. The velocity of the mixture of water and proppants above the proppant bed in the fractures is called the equilibrium mixture velocity (EMV) when the EPH is reached. It is worth noting that the proppant bed will be eroded if the proppant bed height is greater than the EPH. The proppant bed height decreases over time during erosion until the EPH is reached. The Euler–Lagrange method was used by Hu et al. [8] to study the proppant transportation in a single vertical fracture. They suggested that the coarse proppants may be transported first, followed by the fine proppants. The coarse proppants will form a proppant bed quickly and then the fine proppants are transported far in the fractures. Roostaei et al. [9] combined a proppant transport model with the numerical hydraulic fracture model to study the fracturing response and the e ffect of proppant injection on the fracture propagation and dimensions. They found that the fluid viscosity is the most important parameter on the proppant transportation.

Little attention is paid to the amount of proppants entering the secondary fracture, which is a very important quantity for the field engineering. In addition, the mechanism of the proppant moving from the primary fracture into the secondary fracture is also not well understood.

In this paper, the proppant transportation behaviors in the cross fractures are investigated in detail based on the previous work [10] of our group. The Euler–Euler two-phase flow model combined with the kinetic theory of granular flow (KTGF) approach is used. Two dimensionless parameters to describe the proppant distribution in the cross fractures are presented. The first one is the relative EPH, representing the ratio of the EPH to the height of the cross fractures. The other one is the ratio of the proppant mass (RPM) in the secondary fracture to that in the whole cross fracture network. In Section 2, the numerical model and the dimensionless parameters relative to the proppant transportation in the cross fractures are introduced. Compared to the previous work [10], the boundary conditions and the fracture width in the numerical model have been changed to capture more information of the flow behaviors and to be practical for the field engineering. In Section 3, the e ffects of the dimensionless parameters are analyzed. Some suggestions are given for the field engineering application based on the simulated results.
