*2.2. Dimensional Analysis*

All the variables of the problem relating the proppant transportation in the cross fractures based on the equations and the boundary conditions are shown and explained as follows.

The variables of geometry of the cross fractures: the width of the primary fracture and the secondary fracture ( *wa* and *wb*), the length of the primary and the secondary fracture (*La* and *Lb*), the height of the cross fractures ( *H*), the distance from the secondary fracture to the primary fracture entrance (*l*), and the angle between the primary and secondary fracture (bypass angle θ).

The variables of physical properties of the proppant and water: the density of the proppant and the water (ρ*s* and ρ*l*), the average diameter of the proppant (*ds*), and the viscosity of the water (μ*l*).

The variables relating the boundary conditions: the injection velocity of the mixture ( *U*0), the inlet volume fraction of the proppant (<sup>α</sup>*s*0), and the gravity acceleration *g*.

Two parameters to evaluate the quality of hydraulic fracturing are introduced. The first one is the EPH in the primary fracture, and the other one is the RPM. The EPH in the cross fractures is discussed by many researchers [5,13,15,16]. The EPH and the RPM can be written as a causal function of the above variables:

$$f(h, R) = f(\alpha\_{\ast 0}, lL\_0, \varrho; \theta, w\_{\ast}, w\_{\flat \ast}, l\_{\ast \ast}, L\_{\flat \ast}, l, H; \rho\_{\ast \ast}, d\_{\ast \ast}, \rho\_{\mathbb{U}}, \mu\_{\mathbb{U}}) \tag{1}$$

where *h* and *R* are the EPH and the RPM, respectively. The dimensionless causal relationship can be written as [17]:

$$f\left(\frac{\mathrm{h}}{\mathrm{H}},\mathrm{R}\right) = f\left(\alpha\_{s0}, \mathrm{Re}, Ar, \theta, \prime \frac{w\_a}{\mathrm{d}\_s}, \frac{w\_b}{\mathrm{d}\_s}, \frac{L\_a}{\mathrm{d}\_s}, \frac{L\_b}{\mathrm{d}\_s}, \frac{l}{\mathrm{d}\_s}, \frac{H}{\mathrm{d}\_s}, \frac{\rho\_s}{\mathrm{d}\_s}\right) \tag{2}$$

inwhich*h*/*H* is the relative EPH,*Re* = ρ*ldsU*0/μ*l* is the proppant Reynolds number,*Ar* = (ρ*s* − <sup>ρ</sup>*l*)ρ*ld*<sup>3</sup> *s g*/μ<sup>2</sup> *l* is the Archimedes number, *wa*/*ds* and *wb*/*ds* are the relative width of the primary and secondary fracture, *La*/*ds* and *Lb*/*ds* are the relative length of the primary and secondary fracture, *l*/*ds* is the relative distance of the secondary fracture to the primary fracture entrance, *H*/*ds* is the relative height of the cross fractures, and ρ*s*/ρ*<sup>l</sup>* is the ratio of the proppant density to the water density. Here, the relative length and height of the cross fractures as well as the density ratio of the proppant to water are constant. Equation (2) can be rewritten as:

$$f\left(\frac{h}{H}, R\right) = f\left(\alpha\_{\ast 0}, R e\_{\prime} A r\_{\prime}, \partial\_{\prime} \frac{w\_{a}}{d\_{\ast}}, \frac{w\_{b}}{d\_{\ast}}, \frac{l}{d\_{\ast}}\right). \tag{3}$$

The e ffect of the controlling dimensionless independent variables at the left side of Equation (3) on the dependent variables is investigated in this paper. Table A1 in Appendix A shows all the cases adopted in the numerical simulation. The bold part in Table A1 is the standard case. Cases 1–8 are set to study the e ffects of the inlet proppant volume fraction. Cases 3 and 9–13 are set to study the e ffects of the proppant Reynolds number. Cases 3 and 14–18 concern the e ffects of the Archimedes number. Cases 3 and 19–24 concern the e ffects of the bypass angle. Cases 3 and 25–33 are designed to study the e ffects of the relative width of the primary fracture and secondary fracture. Cases 3 and 34–39 are designed to study the relative distance of the secondary fracture to the primary fracture entrance.

The slick-water hydraulic fracturing is widely used in the unconventional resources [3,18]. The slick-water consists of water and chemical ingredients such as the friction reducer, the clay stabilizer, and the bactericide. These chemical additives account for no more than 1% content in slick-water. However, it plays an important role in reducing the friction of the side wall. The viscosity of the slick-water is about 0.8–1.2 mPa·s. Generally, the tap water is used instead of the slick-water in the experiment. In this paper, the tap water is also used with a constant viscosity of 0.001 Pa·s.

#### **3. Results and Discussion**

Figure 2 shows the deposition form of the proppant in the primary fracture at different time points (Case 3). The proppant distribution is similar to the experiment results of previous researchers [1,5,6,19,20]. The proppants first stack at a certain distance from the entrance after entering the primary fracture. The height of proppant bed increases with little change in length until the EPH is reached. Then, the transportation of proppant tends to be stable, as the height of the proppant bed remains unchanged, and the proppant bed only changes in the fracture length direction. The proppant bed at the bottom of the fracture plays an important role in supporting the fractures after the pressure is released. It resembles a stationary porous medium through which the gas and oil will be extracted.

**Figure 2.** The distribution of the proppant in the primary fracture at time (**a**) 10 s, (**b**) 20 s, (**c**) 30 s, and (**d**) 40 s. (Case 3 in Table A1).

In Figure 3a,b, the velocity vector charts of the proppant when the proppant bed height in the primary fracture is about 10% and 75% of the EPH are given (Case 3). The red part of Figure 3 represents the proppant bed. The velocity vector chart of the proppant when the proppant bed moves ahead in primary fracture with the constant bed height is shown in Figure 3c. The proppants settle quickly to the fracture bottom due to the low viscosity of the water initially. As the proppant bed height increases, the following proppants are resisted by the proppant bed and have to move from the upper part of the proppant bed to the depth of the fractures. From the proppant velocity vector chart, it can be found that the proppant bed at the bottom of the fractures does not move. The front part of the bed consists of the following injected proppants, which move forward from the top of the proppant bed surface. Based on the simulation results, the transportation of the proppants in the fractures can be divided into two distinct zones at the steady state: the proppant bed zone and the mixture zone. The mixture zone is the mixture of the proppant and water above the proppant bed.

**Figure 3.** The velocity vector charts of the proppant in the primary fracture, (**a**) the proppant bed height is about 0.1 EPH, (**b**) the proppant bed height is about 0.75 EPH, and (**c**) the proppant bed height has reached the EPH. The red part represents the proppant bed. The length and height of the primary fracture is about 1 m and 0.15 m, respectively. The horizontal coordinates indicate the distance from the entrance of the cross fractures. (Case 3 in Table A1). EPH: equilibrium proppant height.

The inlet proppant volume fraction changes from 1% to 11%. Figure 4 shows the change of the relative EPH (*h*/*H*) with the dimensionless parameters given in Equation (3). The relative EPH increases with the increase of α*s*0 at the value of α*s*0 < 5% (Figure 4a). When the value of α*s*0 is greater than 5%, the relative EPH becomes stable. Although the relative EPH is constant, the time for the proppant bed height to reach the EPH is shorter due to more proppant injection per unit time. For the field engineering application, a larger proppant inlet volume fraction can be used to achieve a faster stability of EPH. However, a large proppant inlet volume fraction may also lead to the blockage of the fractures.

In the field engineering, the sand ratio (ε) is the ratio of the proppant bulk volume to the water volume. The inlet proppant volume fraction (<sup>α</sup>*s*0) is the ratio of the proppant volume to the total volume of the proppant and water. The α*s*0 and the ε values satisfy the relationship α*s*0 = ξε/(ξε + <sup>1</sup>), where ξ is the ratio of the bulk density (ρsb) to the real density (ρ*s*) of the proppant. The sand ratio of the hydraulic fracturing in the field engineering is about 3–8% [21]. Then, the inlet volume fraction of the proppant is about 1–5%. As a result, it can be also concluded that the relative EPH increases with the increase of the sand ratio. According to the simulated results, the most economical sand ratio is about 8%, because the relative EPH becomes stable when the sand ratio is larger than 8%.

The change of the relative EPH with the Reynolds number is shown in Figure 4b. The relative EPH decreases with the increase of the Reynolds number. The Reynolds number characterizes the ratio of the inertia effect to the viscosity effect. A larger Reynolds number causes a higher inertia effect of the proppant and a larger average mixture velocity above the proppant bed. More proppants will be carried far in the fractures. The relative EPH decreases because more proppants on the bed surface move far away in the fractures.

**Figure 4.** The change of the relative EPH with the dimensionless parameters, (**a**) inlet proppant volume fraction, (**b**) proppant Reynolds number, (**c**) Archimedes number, (**d**) angle between primary and secondary fracture (bypass angle), (**e**) relative width of primary fracture, (**f**) relative width of secondary fracture, and (**g**) relative distance of secondary fracture to primary fracture entrance.

Figure 4c,d shows the EPH development with the Archimedes number *Ar* and the bypass angle θ. The results indicate that the relative EPH changes little with the increase of *Ar* and θ. The form of the Archimedes number can be written as:

$$Ar = \frac{(\rho\_s - \rho\_l)\rho\_l d\_s^3 \mathcal{g}}{\mu\_l^2} = \frac{\rho\_l d\_s \mathcal{U}\_s}{\mu\_l} \tag{4}$$

where *Us* = (ρ*s* − <sup>ρ</sup>*l*)*gd*2*s*/μ*<sup>l</sup>* is related to the settling velocity of a single particle in water [22]. As a result, the Archimedes number can also be called as the proppant settling Reynolds number. It represents the settling effect of the proppant, which mainly affects the sedimentation speed of the proppant in the fractures. The proppants quickly settle to the bottom of the fractures due to the low viscosity of the water after injection (Figure 3a). If the settling effect is enhanced, the time for the proppant bed to reach the EPH will be reduced. Table 2 gives the time for reaching the EPH at different values of the Archimedes number. However, the EPH does not change. The change of the relative EPH with the relative width of the primary and secondary fracture as well as the relative distance of the secondary fracture to the primary fracture entrance are shown in Figure 4e–g. With the increase in the values of *wa*/*ds*, *wb*/*ds*, and *l*/*ds*, the relative EPH changes slightly. It can be concluded that the inlet proppant volume fraction and the proppant Reynolds number are the main controlling dimensionless parameters for the relative EPH.

**Table 2.** The time for reaching the EPH at different Archimedes numbers.


The bypass angle, the relative width of the secondary fracture, and the relative distance of the secondary fracture to the primary fracture entrance are the dimensionless parameters related with the secondary fracture. Comparing Figure 4d–g, it is found that the relative EPH is almost constant with the bypass angle, the relative width of the secondary fracture, and the relative distance of the secondary fracture to the primary fracture entrance. In cross fractures, the secondary fracture has little effect on the proppant transportation in the primary fracture. The reason is that the width of the secondary fracture is always small, and the primary fracture is the main channel for the proppant transportation. That means that previous experiments or numerical simulation results in a single fracture can be extend to the cross fractures.

The proppant Reynolds number, which is also called proppant transport Reynolds number, is divided by the proppant settling Reynolds number:

$$
\Pi = \frac{Re}{Ar} = \frac{\mathcal{U}l\_0}{\mathcal{U}\_s} = \frac{\mathcal{U}l\rho\mu l}{d\_s^2(\rho\_s - \rho\_l)g} \tag{5}
$$

where *U*0 is the injection velocity of the mixture. The dimensionless number Π is the ratio of the transport effect to the settling effect. Taking the secondary fracture as a single fracture, the average proppant velocity entering the secondary fracture from the primary fracture is set as the transport velocity. Substituting the parameters into Equation (5), the maximum value of Π in the secondary fracture is about 0.008 in all the cases, which is much smaller than that in the primary fracture listed in Table A1. This means that the settling effect dominates the movement of proppants in the secondary fracture.

An important issue existing in field engineering is how to transport the proppant from the primary fracture into the subsidiary fracture efficiently. The oil and gas flow through the subsidiary fracture into the primary fracture; then, they are collected in the wellbore. Less proppant transported into the subsidiary fractures will cause the blockage of the seepage flow channel of the oil and gas. It is found that there are two mechanisms for the proppant transporting from the primary fracture into

the secondary fracture [6]. The first one is the gravity effect, and the other one is the water-carrying effect. The gravity is along the vertical direction, and it may not drive the proppant movement to other directions directly. The mechanism of the gravity effect may be that the proppants form a high proppant bed in the primary fracture firstly and then enter the secondary fracture under the gravity effect due to the deposition instability. The water-carrying effect is that the drag force on the proppant forms due to the pressure difference between the fracture entrance and the outlet with water entering the secondary fracture at a certain velocity. Figure 5 gives the formation process of the proppant bed in the secondary fracture (Case 3). When the proppants move to the entrance of the secondary fracture, they directly enter the secondary fracture and slowly build up a proppant bed. Hence, it may be inappropriate to use the gravity effects to explain the proppants entering the secondary fracture, and the fluid-carrying effect may be the main controlling factor.

**Figure 5.** The formation process of the proppant bed in the secondary fracture at time (**a**) 10 s, (**b**) 20 s, (**c**) 30 s, (**d**) 40 s, (**e**) 50 s, and (**f**) 60 s. (Case 3 in Table A1).

Figure 6 is a top view of the cross fractures. At the cross section C, the proppants enter from the primary fracture into the secondary fracture. The mass of the proppant in the whole cross fractures and secondary fracture (shaded part in Figure 6) is calculated by the following equation:

$$m\_{\\$} = \iiint\limits\_{V} \alpha\_{\\$} \rho\_{\\$}dV.\tag{6}$$

The RPM can be written as:

$$RPM = \frac{m\_{\text{s,sed}}}{m\_{\text{s,who}}} = \frac{\iiint\_{V\_{\text{swd}}} \alpha\_s \rho\_s dV}{\iiint\_{V\_{\text{who}}} \alpha\_s \rho\_s dV} \tag{7}$$

where *<sup>m</sup>*s,sed and *<sup>m</sup>*s,who are the proppants' mass in the secondary and whole cross fractures, respectively. *V*sed and *V*who are the volume of the secondary fracture and whole cross fractures, respectively.

**Figure 6.** A top view of the cross fractures. The shaded part is the secondary fracture.

Figure 7 gives the change of RPM with the dimensionless parameters mentioned in Equation (7). In order to reflect the distribution of the proppants in the primary fracture and the secondary fracture, the RPM is calculated with the same injected mass of the proppant. The RPM curve with the di fferent inlet proppant volume fraction (<sup>α</sup>*s*0) is shown in Figure 7a. When the inlet volume fraction of the proppant increases, the RPM decreases slightly from 6% to 3%. That means that the increase of the inlet volume fraction does not lead to the increase of the amount of the proppants entering the secondary fracture. More proppants stay in the primary fracture at the large inlet proppant volume fraction. The reason may be that the total resistance of the proppants on the water is larger at the larger inlet proppant volume fraction. The water velocity at the inlet of the secondary fracture is reduced accordingly. As a result, the amount of proppant entering the secondary fracture reduced.

Figure 7b shows the RPM curve with Reynolds number. Similar to the e ffect of the inlet proppant volume fraction, the RPM decreases from 4.6% to 2.5% with the Reynolds number changing from 50 to 300. The proppants hardly enter the secondary fracture when the Reynolds number is large. The RPM changes little with the Archimedes number and the bypass angle (Figure 7c,d). However, the RPM is greatly influenced by the relative width of the primary and the secondary fracture compared to the α*s*0, *Re*, *Ar*, and θ (as shown in Figure 7e,f). When *wa*/*ds* changes from 6 to 11, the RPM decreases from 16% to 3%. This is because when the relative width of the primary fracture is larger, the primary fracture becomes a more favorable channel for the proppant transportation, and more proppants move into the primary fracture, leading to the decrease of the RPM. The RPM increases from 3% to 9% as the value of *wb*/*ds* changes from 2.4 to 5. When the relative width of the secondary fracture is larger, the possession of the primary fracture is weakened, and more proppants move into the secondary fracture, resulting in the increase of the RPM. If the proppants consist of particles of di fferent diameters, the coarse particles will tend to stay in the primary fracture, and the fine particles are accessible to the secondary fracture. Sahai et al. [6] also found that the proppants in the secondary fractures are thinner than those in the primary fracture by using the proppants with certain particle size grading to investigate the sorting e ffect of particles at the intersection of the primary and secondary fracture. In conventional hydraulic fracturing, the naturally ceramsite sand proppants with a certain size grading are used. This may be a good way to improve the hydraulic fracturing and increase the oil and gas recovery. The RPM decreases from 6% to 2.3% with the relative distance of the secondary fracture to the primary fracture entrance increasing from 200 to 1400. This is because the transportation time increases for the proppant entering the secondary fracture when the relative distance of the secondary fracture to the primary fracture entrance increases. When injecting the same mass of proppant, the farther the secondary fracture is from the primary fracture, the less proppant will be transported. It can be concluded that the width of the cross fractures has the greatest impact on the amount of proppant entering the secondary fracture.

**Figure 7.** The curve of ratio of the proppant mass (RPM) with the dimensionless parameters, (**a**) inlet proppant volume fraction, (**b**) proppant Reynolds number, (**c**) Archimedes number, (**d**) angle between primary and secondary fracture (bypass angle), (**e**) relative width of primary fracture, (**f**) relative width of secondary fracture, and (**g**) relative distance of secondary fracture to primary fracture entrance.

In order to make the numerical simulation results practical for field engineering, more cases are simulated, which are shown in Table A1 (Cases 40–65), and the contour map of RPM relating to the dimensionless parameters is given in Figure 8. Each black dot in Figure 8 represents a case in Table A1, and the coordinates indicate the value of the dimensionless parameters. The percentage of the proppant entering the secondary fracture can be estimated in field engineering based on Figure 8.

**Figure 8.** The contour map of RPM relating to the dimensionless parameters, (**a**) inlet proppant volume fraction and the proppant Reynolds number, (**b**) inlet proppant volume fraction and the bypass angle, (**c**) inlet proppant volume fraction and the relative width of the primary fracture, (**d**) inlet proppant volume fraction and the relative width of the secondary fracture.
