# Presentation: 045955

## Coupling Continuum Model and Smoothed Particle Hydrodynamics Methods for Reactive Transport 

**Coupling Continuum Model and Smoothed Particle Hydrodynamics Methods for Reactive Transport **
**Yilin Fang, Timothy D Scheibe and Alexandre M Tartakovsky**
**Pacific Northwest National Laboratory**

**Abstract**

**Constrained Particle Dynamics Approach**

**Objective**

**Continuous Blending Approach**

- Hybrid multiscale simulations are motivated by problems in which large-scale phenomena of interest (e.g., flow and contaminant transport) are strongly influenced by processes occurring at much smaller scales (e.g., diffusive mixing and reactions) that are not well represented by effective or averaged processes or properties. Executing an exhaustive simulation of processes at the smallest scales for a domain of engineering significance is currently impractical. However, the hybrid multiscale approach, in which a small-scale model with high resolution is utilized in a fraction of the overall domain and is linked to a large-scale model with coarse resolution over the remainder of the overall domain, can provide necessary efficiency of characterization and computation that will render solution of these problems practical. Two hybrid model concepts to couple continuum model and smoothed particle hydrodynamics (SPH) methods for reactive transport are presented, one of which uses a mixed interpolation for regions where finite element nodes and particles both have an influence and the overall system is solved together, the other is to use a common overlap domain between the two models and impose the condition that the fluxes of the conserved quantities are continuous across the particle-continuum interface. These approaches combines the advantages of continuum model and SPH. It will allow high resolution simulation around the reaction front using particles and there is no need to refine the continuum domain.

- _[yilin.fang@pnl.gov](mailto:yilin.fang@pnl.gov)_ **·** (509)372-4051
- 3200 Q Ave, K9-36, Richland WA 99352

- Develop and implement an integrated multiscale modeling framework and capability incorporating specific selected process models at continuum scale and discrete pore scale, and apply the hybrid model to simulate complex subsurface flow and biogeochemical transport for subsurface contaminants.

- This approach was proposed by Fernandez-Mendez et al. (Computers and Structures, 83, pp17-18, 2005), targeting the solution of fast-transient dynamics problems, impact simulations, brittle fracture or explosions. In this approach, an arbitrary function u(x) is approximated by:

**Conclusions**
- The coupling approaches require no ramp functions or substitution of nodes by particles.  Particles can be added anywhere they are needed.  They combine advantages of both the efficiency of continuum model and accuracy of discrete SPH model so that problems of concern can be solved at large scales while pore structures at regions of interest in the overall domain are resolved explicitly.

**P** is a vector containing a complete polynomial base of degree less or equal to m.  For 2-D, linear shape function:

*W**a*(x) and ***a* are the weight/shape function and a volume associated to particle x*a.*

- The problem domain is discretized with a Galerkin FEM and particles, numerically integrated using Gauss quadrature in the FE region, and particle integration in the rest of the domain.

- A simple aqueous diffusion example is presented on the right.  Diffusion coefficient is 10-10 m2/s.  2D domain is discretized with 20 FEM nodes, 16 particles, with 4 non-active FEM nodes.  Results at different time are compared to analytical solution, showing a quantitative match.

- Transport equations in FE and SPH domain are solved in Lagrangian-Eulerian formulation to allow larger time steps. Domain of influence for particles is determined by binary search.  Advection will be solved by backward particle tracking approach.

- This coupling scheme has been largely used in coupling continuum scale model with molecular dynamics model (e.g., Europhysics Letters, 52(30), pp271-276, 2000; J. Fluid Mech. 500, pp55-64, 2004).  The approach will be adapted to couple continuum model and SPH model, solving continuum flow and transport model in part of the domain and SPH model in the rest where there is strong coupling between flow and reactions.  An overlap region is used to link these two domains by imposing the continuity of flow and
- transport properties across the interface.

- Boundary conditions on continuum from particles (P→C): continuum quantities are determined from spatial and temporal mean of corresponding particle quantities, for example, velocity and concentration flux at continuum point are:

- Boundary conditions on particles from continuum (C→P):

-  Mass flux continuity across SPH-continuum interface requires particles to be inserted or removed from SPH domain:

- Need to resolve oscillation and stability issue during advective and diffusive transport.

- Constraint on velocity from P→C requires modification of SPH equation of motion:

-  Concentration flux continuity across SPH-continuum interface requires concentrations associated with inserted or removed particles to satisfy: