A new Lagrangian method for transport in porous media (to model chemotaxis in porous media)

As recently shown in laboratory bench scale experiments, chemotaxis, i.e.the movement of microorganisms toward or away from the concentration gradient of a chemical species, could have a fundamental role in the transport of bacteria through saturated porous media. Chemotactic bacteria could enhance bioremediation by directing their own motions to residual contaminants in less conductive zones in aquifers. The aim of the present work is to develop a proper numerical scheme to define and to quantify the magnitude and the role of chemotaxis in the complex groundwater system framework. We present a new class of meshless Lagrangian particle methods based on the Smooth Particle Hydrodinamics (SPH) formulation of Vila & Ben Moussa, combined with a new Weighted Essentially Non-Oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The purpose of this new scheme is to fully exploit the advantages of SPH among traditional meshbased and meshfree schemes and to overcome its inapplicability for modeling chemotaxis in porous media. The key idea is to produce for each particle first a set of high order accurate Moving Least Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a nonlinear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS-WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes. To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions. In the first part, we test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes. In the second part, the new SPH scheme is applied to advection-diffusion equation in heterogeneous porous media with anisotropic diffusion tensor. Several numerical test case shows that the new scheme is accurate. Unlike standard SPH, it reduces the occurrence of negative concentration. In the third part, we show the applicability of the new scheme for modeling chemotaxis in porous media. We test the new scheme against analytical reference solutions. Under the assumption of complete mixing at the Darcy scale, we perform different two-dimensional conservative solute transport simulations under steady-state conditions with instant injection showing that chemotaxis significantly affect the quantification of field-scale mixing processes.