Estimating Drag Forces on Suspended and Laid-on-Seafloor Pipelines Caused by Clay-Rich Submarine Debris Flow Impact

Abstract

Estimating the impact drag forces exerted by a submarine debris flow on a pipeline is a challenge. The conventional geotechnical based methods available to estimate drag forces on buried pipelines in unstable slopes are not applicable to a debris flow impact situation as they ignore or significantly underestimate the shear rate effects in the soil-structure interaction. The results of recent investigations indicate that a fluid dynamics approach in conjunction with rheological principles of non-Newtonian fluids provides a more appropriate way in the study of soil-pipe interaction for submarine debris flow impact situations. To that extent, this paper summarizes the results of a recent investigation on the impact of clay-rich submarine debris flows on suspended (free-span) and laid-on-seafloor pipelines. It presents a method to estimate the drag forces, longitudinal and normal to the pipe axis, for various angles of impact. The investigation comprised experimental flume tests and Computational Fluid Dynamics (CFD) numerical analyses.

Appendix A — Theory for CFD Numerical Analysis

A general description of the theory and the constitutive equations used in the CFD analyses are briefly presented here. In the formulation, the different phases are denoted by lowercase Greek letters, α and β, and the total number of phases is NP. In the inhomogeneous model, each phase has its own velocity and other relevant flow fields while the pressure field is shared between the incompressible fluid phases (CFX 2007). In this model, the fluids interact via the inter-phase mass and momentum transfer terms. The phase continuity equation is expressed by:

$$\frac{\partial}{{\partial t}}\left({r_\alpha \rho _\alpha} \right) + \nabla \bullet \left({r_\alpha \rho _\alpha U_\alpha} \right) = {\rm{S}}_{MS_\alpha} + \sum\limits_{\beta = 1}^{N_P} {\Gamma _{\alpha \beta}} $$

((A.1))

where, r_α, ρ_α, and U_α are the phase volume fraction, density and velocity, respectively, and SMSα is the user specified mass sources. Γ_αβ is the mass flow rate per unit volume from phase β to phase α, which must obey the rule: $\Gamma _{\beta \alpha} = - \Gamma _{\beta \alpha} \Rightarrow \sum\limits_{\alpha = 1}^{N_P} {\Gamma _\alpha = 0.}$

It is important to define the direction of the mass transfer in the conservative equations. A convenient method is to express Γ_αβ by: $\Gamma _{\alpha \beta} = \Gamma _{\alpha \beta}^ + - \Gamma _{\beta \alpha}^ +$. The term Γ_αβ > 0 represents a positive mass flow rate per unit volume from phase β into phase α. The volume fraction is bound by: $\sum\limits_{\alpha = 1}^{N_P} {r_\alpha = 1}$. The momentum equation for a continuous fluid phase is:

$$\begin{array}{l} {\frac{\partial}{{\partial t}}\left({r_\alpha \rho _\alpha U_\alpha} \right) + \nabla \bullet \left({r_\alpha \left({\rho _\alpha U_\alpha \otimes U_\alpha} \right)} \right)} \\ {= - r_\alpha \nabla P_\alpha + \nabla \bullet \left({r_\alpha \mu _\alpha \left({\nabla U_\alpha + \left({\nabla U_\alpha} \right)^T} \right)} \right)} \\ {+ \sum\limits_{\beta = 1}^{N_P} {\left({\Gamma _{\alpha \beta}^ + U_\beta - \Gamma _{\beta \alpha}^ + U_\alpha} \right) + S_{M\alpha} + M_\alpha}} \\ \end{array}$$

((A.2))

where, P_α and μ_α are the pressure and viscosity, respectively, and S_Mα is the user defined momentum sources due to external body forces. T is the matrix transpose operation. M_α is the sum of interfacial forces acting on phase α due to the presence of other phases and is obtained from:

$$M_\alpha = \sum\limits_{\beta \ne \alpha} {M_{\alpha \beta} = M_{\alpha \beta}^D + M_{\alpha \beta}^{LUB} + M_{\alpha \beta}^{VM}} + M_{\alpha \beta}^{TD} + \ldots $$

((A.3))

where, the terms indicated above, in order, represent the inter-phase drag force, lift force, wall lubrication force, virtual mass force and turbulence dispersion force. Finally, the term ($\Gamma _{\alpha \beta}^ + U_\beta - \Gamma _{\beta \alpha}^ + U_\alpha$) represents the momentum transfer induced by the inter-phase mass transfer. The governing transport equations result in 4 × N_P + 1 equations with 5 × N_P unknowns that correspond to (u, v, w, r, P)_α for α = 1 to N_P, where u, v, and w and the velocity components in the x, y and z directions, respectively. Given that the fluids in the inhomogeneous multiphase flow share the same pressure field, the transport equations are solved by imposing the constraint of P_α = P for all α = 1 to N_p