A conceptual model of pore-space blockage in mixed sediments using a new numerical approach, with implications for sediment bed stabilization

Abstract

In mixed sediment beds, erosion resistance can change relative to that of beds composed of a uniform sediment because of varying textural and/or other grain-size parameters, with effects on pore water flow that are difficult to quantify by means of analogue techniques. To overcome this difficulty, a three-dimensional numerical model was developed using a finite difference method (FDM) flow model coupled with a distinct element method (DEM) particle model. The main aim was to investigate, at a high spatial resolution, the physical processes occurring during the initiation of motion of single grains at the sediment–water interface and in the shallow subsurface of simplified sediment beds under different flow velocities. Increasing proportions of very fine sand (D₅₀=0.08 mm) were mixed into a coarse sand matrix (D₅₀=0.6 mm) to simulate mixed sediment beds, starting with a pure coarse sand bed in experiment 1 (0 wt% fines), and proceeding through experiment 2 (6.5 wt% fines), experiment 3 (10.5 wt% fines), and experiment 4 (28.7 wt% fines). All mixed beds were tested for their erosion behavior at predefined flow velocities varying in the range of U _1-5=10–30 cm/s. The experiments show that, with increasing fine content, the smaller particles increasingly fill the spaces between the larger particles. As a consequence, pore water inflow into the sediment is increasingly blocked, i.e., there is a decrease in pore water flow velocity and, hence, in the flow momentum available to entrain particles. These findings are portrayed in a new conceptual model of enhanced sediment bed stabilization.

Appendix

Sediment matrix model

This study uses FLAC-3D to model fluid flow, and PFC-3D to simulate discrete sediment particles. In the discrete element method (DEM), the interaction between particles is considered as a dynamic process achieving a static equilibrium when the mean unbalanced forces are small (cf. Cundall and Strack 1979). The method can serve to model the movement and interaction of assemblies of rigid spherical particles subject to external forces or stresses, such as shear flow in the present case. The particles are placed independently of each other, and interact at common contact points according to chosen contact laws and prescribed boundary conditions—e.g., walls, velocities, or forces. Particles are allowed to overlap each other at contact points. Particle motions are calculated by summation of all inter-particle contact forces (F _N) acting on a single particle, enabling the calculation of the new position of this particle as well as particle displacement using the second Newtonian equation of motion (Cundall and Strack 1979). At the beginning of each time step, the set of contacts is updated from known particle positions. At the contacts the force-displacement law is applied to update the contact forces based on the relative motion between the two entities at the contact and the constitutive model (Itasca 2004a, 2004b).

The relative displacement between two particles at a contact and the contact force acting on the particles are related by the force-displacement law. The contact force is split into a normal component acting in the direction of the normal vector, and a shear component acting in the contact plane. These contact forces can be calculated using the particle overlap U ^N,S (normal and shear):

$$ {F}_{\left(\mathrm{P}\right),\mathrm{N}}={k}_{\left(\mathrm{P}\right),\mathrm{N}}\bullet {U}^{\mathrm{N}} $$

(1)

$$ {F}_{\left(\mathrm{P}\right),\mathrm{S}}={k}_{\left(\mathrm{P}\right),\mathrm{S}}\bullet {U}^{\mathrm{S}} $$

(2)

where k _(P),N and k _(P),S are normal and shear stiffness at the contact, respectively. Computation of particle acceleration and rotation using Newton’s second law yields:

$$ \begin{array}{cc}\hfill {F}_{\left(\mathrm{P}\right),\mathrm{N}\mathrm{e}\mathrm{t}}={m}_{\left(\mathrm{P}\right)}\bullet \overset{..}{x},\hfill & \hfill\ \mathrm{t}\mathrm{ranslational}\ \mathrm{motion}\hfill \end{array} $$

(3)

$$ \begin{array}{cc}\hfill {M}_{\left(\mathrm{P}\right),\mathrm{N}\mathrm{e}\mathrm{t}}={I}_{\left(\mathrm{P}\right)}\bullet \overset{\cdot }{\omega },\hfill & \hfill\ \mathrm{rotational}\ \mathrm{motion}\hfill \end{array} $$

(4)

where m _(P) is the particle mass, I _(P) the moment of inertia, and ẍ and $\overset{\cdot }{\omega }$ the translational and rotational acceleration of a particle, respectively.

Fluid model

The commercial code FLAC-3D (Fast Lagrangian Analysis of Continua) served to simulate fluid flow using a 3D explicit finite difference method (FDM) based on a Lagrangian calculation scheme (cf. ITASCA Consulting Group, Minneapolis, www.itascacg.com). This comprises a three-dimensional grid containing polyhedral elements that represent the material, and is a suitable tool to model nonlinear material behavior such as of soil. Following Wilkins et al. (1963), FLAC-3D enables the representation of three-dimensional bodies and the simulation of their response to applied external or internal forces. It has the ability to simulate fluid flow but, even more importantly, it enables full coupling between a deformable porous solid, made of discrete particles, and a viscous fluid flowing within the pore space. The general discretization of the fluid domain into a discrete body is represented by a finite difference mesh composed of rectangular elements. Equally discretizing a continuous medium by a numerical grid, the DEM differs from the FDM mainly in terms of grid lines and node assembly as well as solution algorithms for partial differential equations within the grid. The fluid domain is discretized into brick-shaped zones defined by eight nodes. An advantage of FLAC-3D is the easy allocation of different fluid-flow properties (isotropic, anisotropic, or null) to distinct zones. Prior to program execution, boundary and initial conditions may be prescribed to a discretized body at given grid nodes and elements. For simplification, and to keep calculation costs low, a highly simplified flow model was chosen, whereby fluid flow was calculated by the FDM approach following Darcy’s law for a homogeneous, isotropic medium and constant fluid density:

$$ {q}_{\mathrm{i}}=-{k}_{\mathrm{i}\mathrm{l}}\widehat{k}(S){\left(P-{p}_{\mathrm{f}}{x}_{\mathrm{j}}{g}_{\mathrm{j}}\right)}_{\mathrm{l}} $$

(5)

where q _i is the specific discharge, P the pore pressure, k the tensor of absolute mobility coefficients of the medium, $\widehat{k}(S)$ the relative mobility coefficient, which is a function of fluid saturation (S), p _f the fluid density, and g _j and subscript l the three-dimensional components of the gravity vector.