Moving Boundaries in Earthscapes Damien T. Kawakami, V.R. Voller, C. Paola, G. Parker, J. B. Swenson NSF-STC

1km Examples of Sediment Fans How does sediment- basement interface evolve Badwater Deathvalley

An Ocean Basin How does shoreline respond to changes in sea level and sediment flux

Sediment mass balance gives Sediment transported and deposited over fan surface by fluvial processes From a momentum balance and drag law it can be shown that the diffusion coefficient is a function of a drag coefficient and the bed shear stress  when flow is channelized = cont. when flow is “sheet flow” diffusion will be non-linear Conic shaped Fan A first order approx. analysis indicates  1/r r radial distance from source Convex shape

An Ocean Basin How does shoreline respond to changes in sea level and sediment flux A large Scale Experiment by Paola and Parker at SAFL has addressed this problem

“Jurassic Tank” ~1m Computer controlled subsidence

How does shore line move in response to sea-level changes Swenson et al can be posed as a generalized Stefan Problem

Base level Measured and Numerical results ( calculated from 1 st principles) Numerical Solution 1-D finite difference deforming grid

The Desert Fan Problem -- A 2D Problem A Stefan problem with zero Latent Heat

A two-dimensional version (experiment) Water tight basin filled with sand –First layer: gravel to allow easy drainage –Second layer: F110 sand with a slope ~5º. Water and sand poured in corner plate Sand type: Sil-Co-Sil at ~45 mm Water feed rate: ~460 cm 3 /min Sediment feed rate: ~37 cm 3 /min

The Numerical Method -Explicit, Fixed Grid, Up wind Finite Difference VOF like scheme Flux out of toe elements =0 Until Sediment height > Downstream basement fill point PE The Toe Treatment Square grid placed on basement At end of each time step Redistribution scheme is required To ensure that no “downstream” covered areas are higher r.05 grid size Determine height at fill Position of toe

Pictures taken every half hour –Toe front recorded Peak height measure every half hour Grid of squares 10cm x 10cm Experimental Measurements

Observations (1) Topography –Conic rather than convex –Slope nearly linear across position and time –bell-curve shaped toe

Observations (2) Three regions of flow –Sheet flow –Large channel flow –Small channel flow Continual bifurcation governed by shear stress

Numerical results Constant diffusion t=360min = 4.7 as a function of t=360min  =170/r, where r=[(iDx) 2 +(jDy) 2 ] 1/2 Diffusion chosen to match toe position

Toe Position constant model

Toe Position r model

Constant r-model Non-Linear Diffusion model shows promise

Moving Boundaries in Earthscapes A number of moving boundary problems in sedimentary geology have been identified. It has been shown that these problems can be posed as Generalized Stefan problems Fixed grid and deforming grid schemes have been shown to produce results in Reasonable agreement with experiments Improvements in model are needed Utilize full range of moving boundary numerical technologies to arrive at a suite of methods with geological application Use large scale general purpose solution packages