1. Basic Research Program Particle-Scale Distribution of Soil Moisture in Porous Media 17 April 2008 Dr. Chris Kees and Dr. Matthew Farthing Coastal and Hydraulics Laboratory

2. Purpose: To provide a multi-scale theoretical and computational model of variably saturated granular/porous media that will improve our ability to perform engineering-scale analyses. Product/Results: Theoretical and computational modeling frameworks for air/water/particle systems. New constitutive relations for macroscopic models of porous media. Payoff: Understanding how the microscopic properties of variably saturated media relate to engineering scale properties will lead to improved ability to detect subsurface features, to build reliable structures, and to predict transport in soils. Schedule & Cost Total $738K MILESTONES Prior FY08 FY09 FY10 Years Army ($K) Other Initial Plan/Prep Particle-Scale Theory Particle-Scale Simulator Three-phase Theory Three-phase Simulator Multiscale Theory Multiscale Simulator Particle-Scale Distribution of Soil Moisture in Porous Materials Status: Basic 6.1 $738K Total Army Program 227 261 250

3. What is the Problem? – We don’t understand the macroscopic structural, hydraulic, thermal, electromagnetic, and chemical properties of variably saturated soils. These properties affect our ability to detect subsurface targets and features, build structures, and predict chemical species transport in soils. What are the barriers to solving the problem? – Accurately measuring thermodynamic conjugate variables in physical experiments under dynamic conditions, as required for formulation of fundamentally sound constitutive relationships, is not possible. Quantities such as phase pressures, surface tension, fluid phase distribution, fluid phase kinetic and potential energies cannot be independently measured at the pore scale. Collaboration across ERDC, commercial firms and/or academia – Ernest Berney (GSL) - Physics of granular media Clint Willson (LSU), - Particle scale measurements Tim Kelley (NCSU) - Multilevel solvers and analysis Masa Prodanovic(UT, Austin) - Pore-scale modeling David DiCarlo(UT, Austin) - Pore-scale theory/experiment Clint Dawson(UT, Austin) - Finite element analysis Graham Carey(UT, Austin) - Finite element analysis Casey Miller(UNC, Chapel Hill) ) - Macroscale modeling High Fidelity Vessel Effects Project (CHL) Countermine phenomenology (GSL) Institute for Maneuverability and Terrain Physics (ITL) What is innovative about this work? The use of particle-scale continuum fluid mechanics simulations that explicitly model the separate phases and the fluid-water and fluid-solid interfaces. The coupling of those models to discrete element models of granular materials to facilitate multi-scale numerical modeling of these systems. What is your publication plan? FY07 - Mini-symposium on near surface air/water flow at U.S. National Congress on Computational Mechanics (July). FY08 - Computer Methods in Applied Mechanics and Engineering (in review). FY09 - Journal of Computational Physics (In preparation), Multiscale Modeling and Simulation (SIAM). Particle-Scale Distribution of Soil Moisture in Porous Materials How will you overcome these barriers? – Apply state-of- the-art computational methods to rigorous continuum thermo-mechanical models of the interaction of air and water phases in granular materials; collaborate with experimentalists and numerical analysis specialists from academia. What are the results of this research and what is its value? – A multiscale theoretical and computational modeling capability for variably saturated granular materials. The ability to calculate macroscopic properties from particle-scale measurements and/or use simulations to supplement experimental methods in complex three- dimensional settings where direct observation of all physical quantities is not possible; a computational multiscale framework that can be used to carry out fundamentally sound engineering analyses.

4. Accomplishments / Status Working 2D/3D air/water Navier-Stokes models based on front-capturing approaches. Working 1D/2D/3D macro-scale (porous media) air/water flow models using locally mass- conservative continuous and discontinuous finite element methods. Added surface integral approximations for jump conditions (e.g. surface tension). Spent five months at UT collaborating with various researchers on numerical and pore- /multi-scale modeling issues.

5. Accomplishments / Status Devised mass/volume conserving correction scheme needed for qualitative correctness, particularly for large surface tension. Implemented initial prototype of adaptive mesh refinement. Added initial support for flux limiters in discontinuous Galerkin finite element methods. Implemented fast-marching and fast-sweeping methods for level set redistancing. Parallelized code by leveraging Argonne’s Parallel Extensible Tookit for Scientific Computing (PETSc). Added higher-order adaptive time stepping by leveraging UNC’s Differential-Algebraic Equation Toolkit (DAETK).

6. Accomplishments / Status Air bubble in water without conservation correction

7. Accomplishments / Status Air bubble in water with conservation correction

8. Accomplishments / Status “Capillary tube” air/water without surface tension

9. Accomplishments / Status “Capillary tube” air/water with surface tension

10. Accomplishments / Status 3D air bubble in water with surface tension (conservative level set method)

11. Accomplishments / Status Water infiltrating into air-dry “cylinder packing”

12. Accomplishments / Status Water infiltrating into air-dry “cylinder packing”

13. Accomplishments / Status Stokes flow of water in a sphere packing

14. Accomplishments / Status Cross-section of a 3D tomographic image showing residual water (blue), air (red), sand (green). Courtesy of Clint Willson, LSU.

15. Accomplishments / Status Currently emphasis is on improving model equations and robustness/flexibility of the implementation. –Surface tension representation in two-phase flows. –Contact line movement in three-phase flows. –Surface tension dominated flows are stiff and require better time integration. –Memory usage, parallelism, computational kernel, and front-end need additional work in order to achieve a multiscale capability.

16. Products Theoretical and computational frameworks for passing information between microscopic two- phase flow/particle systems to macroscopic variably saturated media. New constitutive models for macroscopic models and numerical multiscale models for macroscale engineering models.

17. Technology Transfer High-fidelity vessel effects(Kees and Farthing): Arbitrary Lagrangian-Eulerian mesh technology and porous structure theory/models. Countermine Phenomenology(Howington): Mesh generation capability and moisture distribution theory/models. Stress Transfer in Granular Media(Peters):Discrete Element method coupling and upscaling tools, surface tension theory/models.

18. Publications Non-Conforming Finite Elements for Air/Water Flow in Porous Media. Farthing and Kees. (In preparation 2008) Advances in Water Resources A conservative level-set method for two-phase flow with surface tension effects. Kees, Farthing, Dawson, and Prudhomme. (In preparation 2008) Journal of Computational Physics Locally Conservative, Stabilized Finite Element Methods for Variably-Saturated Flow. Kees, Farthing, and Dawson. (Accepted) Computer Methods in Applied Mechanics and Engineering

19. Publications USNCCM 2007 Minisymposium –Robust Nonlinear Iterative Methods for Time-Dependent Unsaturated Flow. Kees, Farthing, et al. –Locally Conservative, Stabilized Finite Element Methods for Richards’ Equation. Farthing, Kees, et al. –A Computational Tool for Creating Synthetic, Smallscale Infrared Imagery of Vegetated Soil Surfaces. Peters, Ballard, et al. Local Discontinuous Galerkin Approximations to Richards’ Equation. Li, Farthing, et al. Advances in Water Resources 30, 555-575 (2007). Adaptive Local Discontinuous Galerkin Approximations to Richards’ Equation. Li, Farthing, et al. Advances in Water Resources 30, 1883-1901. (2007).

20. Clint Willson (LSU, Civil, Pore scale experiment) David DiCarlo (UT, Petroleum, Pore scale experiment) Graham Carey (UT, ICES, Finite elements) Clint Dawson (UT, ICES, Finite elements) Lea Jenkins (Clemson, Math, Time discretization) John Chrispell (Tulane, Math, Time discretization) Masa Prodanovic (UT, Pore scale modeling) Tim Kelley (NCSU, Math, Solvers) Casey Miller (UNC, Environmental, Macroscale modeling) Collaborations

21. Issues Resources –Need engineers and scientists with UNIX/HPC programming skills –Fast, robust numerical software for parallel architectures (adequate use/understanding of existing tools) Scientific/Technical –Obtaining relevant experimental data and incorporating into the models –Accurately modeling surface tension –Accurately modeling contact line dynamics and statics –Conservation of mechanical energy and mass