As a widely distributed clean and efficient energy source, shale gas has been highly valued by the international energy market and various countries (Peng et al., 2019a). Since the early 21st century, the boom in shale gas development in the United States has significantly advanced the global shale gas development. Since then, the energy landscape in the world has changed gradually (Yuan et al., 2015; Shi et al., 2020). As far as we know, China has abundant shale gas resources and an increasing demand for natural gas. Therefore, a large-scale commercialized development (Ma et al., 2018; Pang, 2018) of hydraulic fracturing shale reservoirs can be expected. Hydraulic fracturing is a necessary technology for developing shale gas (Yu et al., 2020). During the hydraulic fracturing process, the proppant-carrying fracturing fluid is injected into a well at a high pressure and rate to fracture the reservoir rocks. To reduce the friction loss due to tubular roughness, slickwater with drag reducers is often used since it has desirable hydraulic features (Barbot et al., 2013; Shaffer et al., 2013; Al-Muntasheri, 2014; Engle and Rowan, 2014). Numerous studies have focused on various parameters affecting drag reduction (Gallego and Shah, 2009; Chai et al., 2019). Four main parameters can affect the drag reduction of slickwater, including the drag-reducing agent concentration, fluid flow Reynolds number (Re), relative pipe roughness, and water quality (Yang et al., 2019). Meanwhile, two main underlying mechanisms of drag reduction have been identified (Habibpour and Clark, 2017; Habibpour et al., 2017). The first mechanism introduced by Lumley is based on the elongation of coiled polymer molecules, hence increasing the thickness of the viscous sublayer; the other mechanism is the elastic properties of polymers.

Although we have known a lot of the slickwater parameters affecting drag reduction, accurate prediction of drag reduction still remains a challenge. Allahdadi Mehrabadi and Sadeghy (2008) obtained a good drag reduction prediction model called the κ-ε turbulence model for low Reynolds numbers. Al-Sarkhi et al. (2011) developed two correlations to predict the effect of drag-reducing polymers on the friction factor of the two-phase flow for any pipe diameter. Karami and Mowla (2013) presented a general model for predicting drag reduction in crude oil pipelines. Recently, Zhou et al. (2011) proposed a practical pipe diameter model by introducing an effective pipe radius (r _eff) which is defined as follows: r e f f = v f ρ μ , where ρ is the fluid density, μ is the fluid viscosity, and v _f is the flow velocity calculated as (dΔp/4ρl)^1/2. To upscale the laboratory experimental results of the drag reduction of slickwater to the field application, two conditions must be met: first, the r _eff value of the field pipe (tubing/casing) must be in the same range as the laboratory pipe; second, the slope of r _eff versus v of the field must match the laboratory setup, where v is the average fluid velocity. We found that as long as the slopes of effective pipe radius versus velocity from the laboratory pipe and the field pipe are numerically close, field drag reduction can be predicted by a modified correlation between DR and velocity established in the laboratory (Zhao et al., 2018; Chen et al., 2021). In this study, two novel general models are proposed using the Reynolds criterion to simplify the prediction condition.

The models presented in this study are based on the similarity principle so as to upscale the laboratory experimental results to the field application. Due to laboratory setting constraints that are different from those in the field application, the experimental results can be upscaled to the field application only under some specific conditions. The laboratory measurements can accurately represent the flow dynamics of the corresponding prototype. Therefore, the similarity between the model and the prototype must be satisfied. There are three types of similarity principles: geometric similarity, kinematic similarity, and dynamic similarity. It is relatively easy to achieve the geometric and kinematic similarities, while the dynamic similarity requires an equal ratio of forces acting on the two systems. Usually, dynamic similarity can be achieved by equating such flow dynamic dimensionless variables as Froude number and Reynolds number. In this study, we chose the Reynolds number as the inherent parameter because it can be easily measured in both the laboratory condition and the field application condition. Reynolds number (Re) is a dimensionless number that is commonly used to characterize flow patterns in different fluid flow situations (Peng et al., 2019b). Because of the similarity of flow viscosity between the prototype and the model, the Reynolds number of the field fracturing flow model should be equal to that of the lab drag reduction experiment model, according to the similarity of flow dynamics.

In previous studies, we have developed a modified Virk’s correlation to accurately characterize the friction factor (λ) under different Reynolds numbers in the turbulent flow pattern for the polyacrylamide polymer drag-reducing agent using laboratory data. The unified model is shown in Eq. 1. 1 λ = α D β lg ( Re λ ) + γ D θ . (1)

This equation can be changed to Eq. 2. Re l a b = a D l a b m λ l a b n , (2) where Re_lab is the Reynolds number based on the laboratory data, D_lab is the lab pipe diameter, and λ_lab is the friction factor from the laboratory data. For a given pipe diameter in the field, a similar relationship between the field Re (Re_fie) and the field λ (λ_fie) can be obtained by Eq. 3. Re f i e = b λ f i e n . (3)

The Reynolds numbers of the lab and field can be equated by Eq. 4: λ f i e = ( a D l a b m b ) 1 / n λ l a b . (4)

For a slickwater solution, the λ_lab and λ_fie can be determined by the Hagen–Poiseuille equation, respectively, as follows: λ l a b = π 2 D l a b 5 Δ P l a b 8 ρ L l a b Q l a b 2 , (5) λ f i e = π 2 D f i e 5 Δ P f i e 8 ρ L f i e Q f i e 2 , (6) where ∆P _lab is the pressure drop in the laboratory experiment, ρ is the density of slickwater, L _lab is the pipe length in the laboratory experiment, Q _lab is the volumetric flow rate in the laboratory experiment, ∆P _sie is the field friction of slickwater, L _fie is the well depth, and Q _fie is the field volumetric flow rate.

Eq. 5 and 6 are substituted into Eq. 4 to get the predicted ∆P _fie by Eq. 7: Δ P f i e = ( a b D l a b m ) 1 / n D l a b 5 Δ P l a b L f i e Q f i e 2 L l a b Q l a b 2 D f i e 5 . (7)

For the given L_fie, Q_fie, and D_fie, ∆P _fie can be calculated according to the process shown in Figure 1.

The field friction of pure water (∆P _pw ) can be calculated by Eq. 8, which is obtained by the friction-gradient flow rate diagram of water, as shown in Supplementary Figure S1 in Supporting Information. Δ P p w = 1.385 × 10 6 D f i e − 4.8 Q f i e 1.8 L f i e . (8)

The drag reduction (DR) is defined as follows: D R = 100 - Δ P f i e Δ P p w × 100. (9)

Eq. 7 and 8 are substituted into Eq. 9 to get the DR: D R = 100 - 7.22 × 10 - 5 ( a b D l a b m ) 1 / n D l a b 5 Δ P l a b Q f i e 0.2 L l a b Q l a b 2 D f i e − 9.8 × 100. (10)

Two novel general models are established to predict the field drag reduction using laboratory experimental data based on the dynamic similarity theory. When the laboratory and field Reynolds numbers are the same and the laboratory and field volumetric flow rates and pipe diameters are known, general model 1 can be used. When the field volumetric flow rates and the laboratory pipe diameter are known, general model 2 can be used. The two proposed mathematical models for field drag reduction prediction are validated by 42 data points. The validation results showed that both models can give high-accuracy predictions, with a deviation ranging from −10 to 10% using general model 1 and a deviation ranging from −6 to 6% using general model 2.