# MPM-Flow project -- Description

## Problem and target

During a flow slide large amounts of soil move downslope within seconds or gradually over hours; on the scale of river shores to continental slopes. These can be triggered in loose sand by static or dynamic liquefaction induced by erosion and sedimentation, scour and earthquakes including induced earthquakes [DS04, DT01, MW79]. A small increase in pore pressure, for example through a shockwave, suffices to fluidise a loose sand body. Alternatively, in denser

sand, flow slides might be initiated through breaching [Mastbergen, van den Berg]. A local steepening of a slope, caused for example by scour, forms the initial situation for a flow slide. With water gradually sucked into sand pores, sand fluidises in thin layers along this local front. With sand flowing downwards, the ‘step’ thereby moves upwards along the slope. The downslope flow of fluidised soil might further erode the soil surface picking up more and more soil, increasing in speed.

For the accurate prediction of flow slides the exchange between pore water inside the soil and (turbulent) free-surface water flowing along the soil surface has to be considered as well as the entrainment of water in a moving soil mass. furthermore, soil is rarely homogeneous as assumed in simplified models or laboratory experiments. In reality it exhibits interlayering and inter-bedding at the scale of the representative operating processes. Such small and large scale spatial variation in e.g. packing density and permeability, influence the onset of flow slides [JT14]. They also influence the creation of erosion channels inside a slope and the generation and dissipation of pore water pressures before, during and after onset of a flow slide. Likewise, their impact on the transport of inhomogeneous sediments has to be taken into account. An initially inhomogeneous fluidised sand-water mixture, for example in the form of lumps of soil carried by water current, can be transformed through turbulences into a homogeneous mix of sand moving further downslope by gravity [MC14, JE12a, JE12b]. This requires tracing density and viscosity of the fluidised soil. Existing empirical solutions on erosion and sedimentation and partial soil or fluid mechanical treatments fall short in sufficiently capturing these interacting processes.

Much progress has been made throughout the last decades in numerical analyses of geotechnical problems including problems that involve groundwater flow, for example by the Plaxis 3D FEM software. Simulations of fluid flow in hydraulic engineering applications based on solution of the Navier-Stokes equations reached a high level of sophistication, for example with the Delft3D computational fluid dynamics (CFD) software. However, no integrated solution exists to date for the combined modelling of deformation of water-saturated soil and flow of free surface water, the transition between the two. Soil-water interaction, i.e. erosion and sedimentation, is currently modelled with available software on the basis of empirical relations rather than a consistent continuum mechanical description [HV06].

A solution which is based on interfacing geotechnical engineering and CFD software is not straight forward. Geomechanical problems require a Lagrangian description and the widely used finite element method (FEM) is commonly used for their solution. CFD software follows an Eulerian approach and commonly uses a Finite Difference scheme. Combining such differing approaches renders numerical inaccuracies. In the current project, a novel integrated numerical solution for the analyses of underwater flow slides from initiation up to deposition of sediments will be developed on the basis of present numerical state-of-research approaches.

Throughout the last years considerable progress has been made in numerical analyses of geotechnical problems involving large deformations of water-saturated soil by means of the Material Point Method (MPM) [LB11, IJ12]. The MPM is closely related to the FEM. It combines the Lagrangian approach of the FEM with the Eulerian approach of particle methods such as SPH (smoothed particle hydrodynamics) [HB07]. Equilibrium equations are solved on a background finite element mesh as with the FEM. A cloud of material points that moves through the mesh is used to model arbitrary large deformations of soil, or flow of water. Mass conservation is implicitly obeyed. A separation of material, gapping or erosion-like processes is implicitly included in this mixed Lagrangian-Eulerian approach. It furthermore features a straightforward soil-structure and water-structure contact formulation. Several highly non-linear density dependent strain softening models are available. They are well suited to model sand. The MPM has been extended for coupled 2-phase analyses. Recently, it was found that this numerical method is well suited to simulate problems of erosion and sediment transport [ZW13]. Soil with water flowing through its pores, fluidized soil and the transitions between the two states are modelled in an integral numerical framework. Such an MPM suited for first simulations of flow slides of homogeneous sediments is currently developed at Deltares together with University of Cambridge, UPC Barcelona, TU Hamburg-Harburg, TU Delft and University of Pdadova. This MPM Software will be used in this project for the numerical analyses of flow slides and other problems of erosion. This however requires significant enhancement of the MPM code through integration of existing physics based models and new models developed in the course of the project.

*Figure 1: MPM simulation of slope deformation [ZW13] showing soil “particles” in the presence of fluid.*

## Methodology

In order to gain a deeper understanding of flow slides and to develop an integrated numerical solution based on MPM the following straightforward research methodology has been drawn up:

- experiments in laboratory and field;
- physics based modelling;
- mathematical / numerical modelling;
- development of integrated software;
- validation of numerical solution.

This methodology results in six workpackages as described in detail below.

#### Work package 1: Laboratory experiments

A set of key laboratory experiments will be performed to deepen the understanding of flow slides in an integral (interdisciplinary) manner. Here, the experimental boundary conditions are extended into a natural heterogeneous context. Fluid pressures in pores and surrounding water bodies, deformation rates of soil, flow rates of turbulent suspensions and evolution of slope topography are monitored during flow slides. Initial soil composition and heterogeneity, topography, density and porosity are controlled boundary conditions. Flow slides will be triggered by 1) undercutting, 2) fluid injection, 3) over-steepening in analogy to natural and man-made triggers for flow slides. Procedures for these experiments can be readily developed from previous work in the UU Eurotank Flume Laboratory. Smaller scale laboratory experiments will be performed at the Laboratory Fluid Mechanics at DUT with a stronger focus on measuring the water motion.

Experiments will be supplemented by studies provided by industrial partners. Furthermore, data of an elaborate field test in the Western Scheldt that is set up by ‘Stichting IJkdijk’ with support from the Dutch Ministry of Infrastructure and the nvironment in collaboration with Deltares and other partners of industry will be used. It is scheduled to take place in the second half of 2014. It involves the triggering of a full-scale flow slide, recorded through a comprehensive measurement system. The integration of such a field experiment with laboratory experiments ensures that scale comparison and scaling are engrained in the project.

#### Work package 2: Validation MPM

Extensive validation of the MPM software will be performed in the course of this project. The planned simulations of laboratory and field tests as well as case studies provided by industry will be highly complex, requiring good understanding of the numerical method and the considered problem. As a starting point, simplified benchmark problems based on analytical solutions will be used for validation. Here, the available Deltares MPM code can be used. As the enhancement of the MPM proceeds, the complexity of analyses will be gradually increased from laminar to turbulent flow, from homogeneous to heterogeneous soil, from the available simplified state transition criteria to more sophisticated solutions.

#### Work package 3: Modelling soil-water interaction / soil heterogeneity

Results obtained from experimental studies will be translated into physics based models of soil heterogeneity and soil-water interaction, i.e. erosion, transport and deposition of sediments, possibly based on existing solutions. Soil modelling is not an objective of this project. Sophisticated constitutive models for sand are available with the MPM. If necessary, they might be extended.

#### Work package 4: Modelling flow and turbulence

Currently, only laminar flow of free-surface water is considered with the MPM. However, a proper description of the turbulence dynamics is necessary as it governs local shear and normal stresses as well as their fluctuations. A first step is the introduction of the so-called k-e turbulence model in order to include the statistical mean turbulence properties. In a second stage, the Large Eddy Simulation technique will be used to account for spatial and temporal variability of the stresses exerted by the flow. Penetration of pressure fluctuations and entrainment of soil particles and parcels is affected by the flow heterogeneity whereas local soil motion drives the flow and its fluctuations. Linking these mutual interactions without introducing instabilities or numerical artefacts is one of the challenges of this research. Furthermore, the fate of the transported soil is highly affected by turbulence.

#### Work package 5: Numerical solutions

Anticipated long timeframes and large dimensions of flow slide analyses as well as the complexity of the considered equilibrium and transport equations requires highly advanced, purpose-built numerical solutions in order to obtain usable, i.e. stable, software allowing for analyses whose computational effort is affordable and computation time remains within a reasonable timeframe.

The numerical integration of the dynamic equilibrium and transport equations over time involves advancement of the solution in steps. Depending on the differential equation and time integration scheme, different constraints are placed on the time increments to ensure stable solution. For propagation of waves in soil described by hyperbolic differential equations a different criterion applies than for dissipation of pore water out of soil, described by a parabolic equation. For implicit and explicit schemes different parameters of a problem are known to limit the step size, e.g. stiffness of soil. The impact of other parameters such as soil permeability is not well understood yet.

In principle two methods are used for the analysis of the stability of time stepping methods: 1) methods based on the amplification factor of the integration method combined with an estimate of the eigenvalue of the corresponding (linearized) space discretisation method or 2) a Fourier analysis. Gained insights can be used to invent more stable integration methods [SP05, SP08]. Another problem lies in the transition between soil filled with pore water whose mechanical behaviour is described by equilibrium equations of the soil-water mixture and fluidized soil described by the Navier-Stokes equations along a moving interface. This is presently modelled in a simplified, discretised way. State transition is detected on the basis of a threshold porosity of the soil. Here, an accurate, robust solution has to be developed that takes into account a gradual transition between the two states.

#### Work package 6: Integration of results into MPM

Integration of results into the Deltares MPM involves extension of the MPM code or development of libraries which are linked to the MPM code. Special attention will be paid to efficient parallelisation of the code to be able to make optimal use of available state-of-the-art computer facilities. Sets of benchmark tests will be assembled for respective code extensions to prove their proper working. Extensions will be thoroughly documented and coding guidelines will be applied to ensure a high degree of quality of the implemented solution.