CFD Methodology to Determine the Hydrodynamic Roughness of a Surface with Application to Viscous Oil Coatings

Known reserves of nonconventional oils represent one of the largest global petroleum resources (Nunez et al. 1998; Saniere et al. 2004; CAPP 2015). These reserves mostly comprise asphaltic, dense, and very viscous oils, such as bitumen and heavy oil. The production and transportation of these nonconventional oils require extraordinary techniques that are not needed for conventional petroleum deposits. Various mining and in situ production technologies are used to extract these nonconventional oils. After extraction, these viscous oils are transported from a company’s various production sites to its upgrading facilities (Joseph et al. 1999; Sanders et al. 2004). Pipelines are most commonly used for this transportation, provided the effective viscosity of the oil can be reduced through heating, dilution, partial upgrading, or emulsification (Nunez et al. 1998; Saniere et al. 2004; Jean et al. 2005). These methods require substantial capital investments and operating costs, which contribute significantly to the higher production costs of heavy oil. As a result, the oil industry is keen to use more cost-effective technology for heavy oil transportation (Nunez et al. 1998; Guevara et al. 1998; Saniere et al. 2004; Hart 2014).

Water-assisted pipeline transportation is a promising alternative technology for transporting viscous oils like heavy oil and bitumen. Here, viscous oil flows in the core and water flows through the annulus. The annular water film mostly protects the viscous oil from touching the pipe wall and thereby acts as a lubricant. The lubricating water is either injected or already present in the mixture (Arney et al. 1993; Joseph et al. 1999; McKibben et al. 2000; Sanders et al. 2004; M. McKibben, R. Gillies, S. Rushd, and S. Sanders, “A new model to predict pressure losses in the water-assisted pipeline transport of bitumen or heavy oil,” submitted, J. Petrol Explor. Prod. Technol., Springer, Berlin). This pipeline transportation technology is referred to as lubricated pipe flow (LPF). It requires much lower energy input compared with the transportation of viscous oil alone in the pipeline (Rodriguez et al. 2009; Crivelaro et al. 2009).

A concern for the application of LPF is that some oil tends to permanently adhere to the pipe wall (Saniere et al. 2004). This phenomenon is called wall fouling. The primary causes of wall fouling appear to be through direct contact of the oil core with the wall generally because of a low water cut (i.e., a thin lubricating layer), low operating velocities such that the hydrodynamic lift forces are not strong enough to keep the core away from the wall, and density differences between the core and the annular fluids (Arney et al. 1996; McKibben et al. 2000; Schaan et al. 2002; Rodriguez et al. 2009). Other contributors to wall fouling include pipe material (wettability), pipe shutdowns (i.e., no-flow conditions), and transport of oil droplets from the annulus to the wall (Arney et al. 1993; Joseph et al. 1999; Sanders et al. 2004). Even though frictional pressure loss in a fouled pipe is higher compared with that for similar transportation in an unfouled pipe, the loss is substantially lower than what would be expected for transporting only heavy oil or bitumen (Arney et al. 1996; Joseph et al. 1999; McKibben et al. 2000; Schaan et al. 2002; Sanders et al. 2004; Rodriguez et al. 2009; Crivelaro et al. 2009; M. McKibben, R. Gillies, S. Rushd, and S. Sanders, “A new model to predict pressure losses in the water-assisted pipeline transport of bitumen or heavy oil,” submitted, J. Petrol Explor. Prod. Technol., Springer, Berlin). It should be noted that wall fouling does not appear to destabilize the annular (lubricated) flow regime even though it produces higher friction losses (Joseph et al. 1999; Rodriguez et al. 2009; M. McKibben, R. Gillies, S. Rushd, and S. Sanders, “A new model to predict pressure losses in the water-assisted pipeline transport of bitumen or heavy oil,” submitted, J. Petrol Explor. Prod. Technol., Springer, Berlin).

Formation of a wall-fouling layer of oil is practically unavoidable in the industrial-scale applications of LPF technology (McKibben et al. 2000; Schaan et al. 2002; Shook et al. 2002; Rodriguez et al. 2009). Different degrees of wall fouling are experienced in various applications of LPF, making it possible to divide LPF into two major categories depending on the extent of fouling: core annular flow (CAF) and continuous water-assisted flow (CWAF). Core annular flow primarily denotes a somewhat idealized concept of LPF because it involves a core of viscous oil lubricated with a water annulus in an unfouled pipe (Arney et al. 1993). In practice, most (if not all) commercial applications of LPF can be categorized as CWAF.

Presently, a reliable model to predict pressure losses in CWAF is not available (Shook et al. 2002; Hart 2014; M. McKibben, R. Gillies, S. Rushd, and S. Sanders, “A new model to predict pressure losses in the water-assisted pipeline transport of bitumen or heavy oil,” submitted, J. Petrol Explor. Prod. Technol., Springer, Berlin). Although numerous empirical, semimechanistic, and numerical models have been proposed (e.g., Arney et al. 1993; Ho and Li 1994; Joseph et al. 1999; Rodriguez et al. 2009; Crivelaro et al. 2009), all are very limited in applicability. Some are only appropriate for CAF, and others are highly system-specific. None of the existing models explicitly addresses the effect of wall fouling on frictional pressure losses.

Although the layer is relatively thin (compared with the pipe diameter), it is textured or rippled, which can significantly increase hydrodynamic roughness (Brauer 1963; Picologlou et al. 1980; Shook et al. 2002). The rippling effect (i.e., the irregular waves that form on the viscous coating layer) is associated with interfacial instability due to viscosity stratification (Hooper and Boyd 1987; Kushnir et al. 2014). Such instability is a result of the interaction of the two fluids through the velocity and viscous stresses at the interface (Tilley et al. 1994; Govindarajan and Sahu 2014). The instability effect is discussed extensively in the literature, but it is not the focus of this study; instead, the goal here is to evaluate the effect of the rippled coating on hydrodynamic roughness.

The engineering scale for hydrodynamic roughness is the Nikuradse sand grain equivalent (Flack and Schultz 2010). This kind of equivalent roughness is used extensively for commercial metal pipes or channels. Similar roughness is also used for various atypical rough walls, such as metal walls with nonuniform roughness, minichannels, or biofilms on a solid wall (Picologlou et al. 1980; Bayazit 2010; Kandlikar et al. 2005). A biofilm is somewhat similar to a viscous oil coating in that it is conformable and can substantially increase hydrodynamic roughness (Picologlou et al. 1980; Barton et al. 2005; Schultz 2007; Lambert et al. 2009; Andrewartha 2010). Although some measurements of roughness caused by a biofilm are available in the literature, nothing similar can be found in the literature for viscous oil coatings.

The objective of the present study, therefore, is twofold. First, a methodology based on computational fluid dynamics (CFD) is tested that would allow for the determination of hydrodynamic roughness under conditions where more conventional methods are difficult to implement or simply unsuitable. For example, velocity profile measurements can be used to infer the hydrodynamic roughness of a surface. In some cases, these cannot be made, usually for economic or technical reasons. In other cases they can be made but, because of flow cell aspect ratio, the measured velocity profile cannot be easily related to the hydrodynamic roughness owing to the presence of strong secondary flows (e.g., Melling and Whitelaw 1976). Then there are cases where the CFD-based methodology can be used instead of standard correlations. For example, the Colebrook formula cannot be used under conditions involving rough-wall asymmetry—such as, two channel walls of different roughness and two smooth walls.

Additionally, hydrodynamic roughness produced by a very viscous coating layer, which is an idealized version of the fouling layer in LPF, is evaluated. The broader motivation, as mentioned previously, is that existing models of LPF do not explicitly address the effect of this fouling layer, even though its contribution to overall friction loss is known to be significant.

The CFD-based approach for calculation of hydrodynamic roughness was validated by applying it to analyze data obtained from three independent sets of experiments: clean flow cell tests, sandpaper tests, and tests where surface roughness was produced by a biofouling layer. The experiments involved both solid and soft rough surfaces, which produced equivalent sand grain roughness values in the range of 0–5 mm. Each test case is discussed in detail in the following sections.

In this study, a CFD-based approach was used to determine the hydrodynamic roughness of a rough surface based on bulk measurements—flow rate and frictional pressure loss. Typically, velocity profiles must be measured above the rough surface. This approach is known to be time-consuming and, for a more complicated flow, such as lubricated pipe flow, would be technically infeasible. Another alternative is the use of friction factor correlations, such as the Colebrook-White equation, but this approach can provide very large changes in the value of $k_s$ for relatively small changes in the measured pressure loss. Therefore, the CFD-based approach described here is viewed as a viable alternative. Compared with the approach where velocity profiles are measured to determine $k_s$, the CFD approach is extremely time- and cost-effective. The CFD approach was tested against the velocity profile method applied to determine the hydrodynamic roughness of a biofouled surface. Good agreement was obtained. Subsequently, the CFD methodology was applied to determine the hydrodynamic roughness of two different sandpapers. Here, the CFD results were compared with those obtained using the correlation of Flack and Schultz (2010), which uses the statistics of the surface topology. Again, good agreement was observed between the two methods. The performance of the CFD methodology is highly dependent on the selection of a turbulence model. Here, the 7-equation $\omega$-RSM model, which was developed for anisotropic flow conditions and/or flow over rough surfaces, was shown to be superior to the two-equation $k\text{-}\omega$ model for the range of conditions studied. Although the CFD methodology provided reasonable predictions here, other turbulence models should be tested, particularly for very high Reynolds number conditions.

The other primary focus of this study was to gain new understanding of the hydrodynamic behavior of a viscous oil coating, given that such coatings are a dominant mechanism for pressure loss in LPF. The present study was intended to provide some preliminary information about the relationships among coating thickness, flow Reynolds number, and hydrodynamic roughness. Additionally, the surface topology of the viscous oil coatings was measured. In the rectangular flow cell tested here, it was discovered that the hydrodynamic roughness was nearly independent of $\mathsf{R}$ (for ${10}^4<\mathsf{R}<{10}^5$) and approximately proportional to the coating thickness. Similar findings were observed when the coated-wall pipe flow data of Brauer (1963) and Shook et al. (2002) were reanalyzed, giving confidence that the rectangular flow cell measurements made here are consistent with pipeline flow behavior. One important limitation is that, if the oil coating is partially stripped from the wall, these results will not hold. Therefore, additional tests must be done where oils with different viscosities are tested. Additionally, larger pipes (i.e., higher Reynolds numbers) must be tested. Also, before the results of this study can be applied with confidence to LPF applications, the impact of oil droplet interaction between the mean flow and the coated wall should be investigated. This could be accomplished by conducting experiments similar to those described here, except that the flowing fluid should consist of oil droplets suspended in water.

Finally, this is the first time, to the authors’ knowledge, that the Flack and Schultz (2010) correlation has been applied to viscous oil coatings. The surface topology measurements made here, and the use of the Flack and Schultz correlation to convert those measurements to a value of hydrodynamic roughness, proved to be very valuable. It was determined that, for a given coating thickness, the amplitude of the surface roughness did not change substantially with Reynolds number; however, as the Reynolds number increased, the roughness became more structured. Again, it would be most useful to determine in the future how oil viscosity affects this relationship.

The following symbols are used in this paper:

$A$

cross-sectional area perpendicular to flow direction (${\mathrm{m}}^2$);

$A_p$

surface area on test plate (${\mathrm{m}}^2$);

$D$

diameter (m);

$D_h$

hydraulic diameter (m);

$D_{ij}$

a tensor;

$h$

height of flow domain (m);

$k$

turbulent kinetic energy ($\mathrm{J}/\mathrm{kg}$);

$k_s$

Nikuradse sand grain equivalent roughness (m);

$l$

length of flow domain (mm);

$m_w$

mass flow rate of water ($\mathrm{kg}/\mathrm{s}$);

$P_{ij}$

Reynolds stress production tensor;

$p$

static (thermodynamic) pressure;

$\mathsf{R}$

Reynolds number;

$R_a$

average roughness (m);

$R_{\text{RMS}}$

root mean square roughness (m);

$R_{sk}$

skewness of roughness;

$R_z$

peak-to-valley average roughness (m);

$S_i$

sum of body forces (N);

$T$

temperature (°C);

$t_c$

average thickness of wall-coating layer (m);

$U_i$

velocity vector ($\mathrm{m}/\mathrm{s}$);

$u_{\tau }$

friction/shear velocity ($\mathrm{m}/\mathrm{s}$);

$u^{+}$

dimensionless velocity;

$V$

average velocity ($\mathrm{m}/\mathrm{s}$);

$W$

width of flow domain (m);

$y^{+}$

dimensionless distance from wall;

${\delta }_{ij}$

identity matrix or Kronecker delta function;

$\mathrm{\Delta }P$

pressure drop (Pa);

$\mathrm{\Delta }P/L$

pressure gradient ($\mathrm{Pa}/\mathrm{m}$);

$\mu$

viscosity ($\mathrm{Pa}·\mathrm{s}$);

${\mu }_o$

viscosity of coating oil ($\mathrm{Pa}·\mathrm{s}$);

${\mu }_t$

turbulent eddy viscosity ($\mathrm{Pa}·\mathrm{s}$);

$\rho$

density ($\mathrm{kg}/{\mathrm{m}}^3$);

${\rho }_o$

density of oil ($\mathrm{kg}/{\mathrm{m}}^3$);

${\tau }_{ij}$

stress tensor;

${\mathrm{\varphi }}_{ij}$

pressure-strain tensor; and

$\omega$

turbulent eddy frequency ($1/\mathrm{s}$).

This research was conducted with support from the NSERC Industrial Research Chair in Pipeline Transport Processes (RSS). The authors acknowledge the support of Canada’s Natural Sciences and Engineering Research Council (NSERC) and the sponsoring companies: Canadian Natural Resources Ltd., CNOOC-Nexen Inc., Saskatchewan Research Council’s Pipe Flow Technology Centre, Shell Canada Energy, Suncor Energy, Syncrude Canada Ltd., Total, Teck Resources Ltd., and Paterson & Cooke Consulting Engineers Ltd.