Code Development and Numerical Simulations

Advection-Diffusion-SPH (Kordilla et al. 2014) is a discretization of the fully coupled Landau-Lifshitz-Navier-Stokes (LLNS) and stochastic advection-diffusion equations. It allows the simulation of diffusion and advective transport processes on arbitrary macroscopic (Fickian diffusion) to microscopic (atomistic) scales where the hydrodynamic behavior begins to be dominated by thermal fluctuations governed by the fluctuation-dissipation theorem. The additional random forces are added to the momentum-conservation equation following the classical SPH formalism in symmetric form:

\( \mathbf{F}_i^{\xi} = \sum_{j=1}^{N}\left(\frac{\boldsymbol{s}_j}{n_{j}^2}+\frac{\boldsymbol{s}_i}{n_{i}^2}\right)\cdot \frac{\mathbf{r}_{ij}}{r_{ij}}\frac{ d W(r_{ij},h)}{d r_{ij}} \)

where the magnitude of the random stress is scale- and timestep dependent and the lm-component of the stress tensor is given by

\( s^{lm}_i = \sqrt{ \frac {4 h^3 \mu \widetilde{k}_B T \delta^{lm} n_i^2}{\Delta t } } \xi_i^{lm} \)

with a unit-variance random number \( \xi \). Our model is able to properly recover all fundamental spatial statistics induced by fluctuations and associated divergence of the interface power spectrum and its scale-invariant nature of the perturbed surfaces. Despite the microscopic nature it is remarkable that gravitational effects are strong enough to dampen the evolution of the interfaces.

We are currently involved in the modeling of karst and fractured aquifers, amongst others in Israel, France and India. Due to limitations in computational capacities and/or data provision it is often not feasible to explicitly resolve individual fractures, karst conduits or other highly conductive pathways in distributive modeling approaches. Nevertheless, while the exact location and geometry of such features is often difficult to detect, they may clearly dominate the shape of spring signals, control the aquifer response of hydraulic tests and can be recognized in tracer studies and associated arrival times.

Dual-continuum (multi-continuum) models rely on a decomposition of the model into a set of hydraulically interacting domains \(d=1,…,n\), each with a volumetric fraction of the total porosity \(w_d\), specific hydraulic properties (flux \(q_d\), storage terms \(S\)) and coupled by a suitable exchange coefficient \(\Gamma_{ex}\).

\( -\nabla w_d(q_d) + \Gamma^{ex} \pm R_d = w_d \frac{\delta \theta_d^s S_d^w}{\delta t} \)

This allows modeling of complex aquifer dynamics, including the karst-specific rapid system reaction followed by a slower porous matrix dominated recession period. Transport dynamics, such as environmental tracer routing or contamination simulations, which often exhibit a clear coupling between porous matrix and the highly conductive  fracture system, can easily be modeled as well.

Furthermore we are interested in the coupling of hydrological surface flow and vadose recharge dynamics, specifically in (semi-)arid regions where the limited precipitation volumes are often highly concentrated (spatially and temporally) and trigger a variety of challenging processes during the routing from surface to water table.

Discrete fracture network or fault zone models commonly employ a classical parallel plate description based on the Navier-Stokes equation to model aperture-dependent fluid flow within highly conductive elements embedded in a porous matrix:

\( Q = -\frac{\rho g}{12\mu} \frac{\Delta h}{\Delta x} (2h)^2 A \)

Compared to discrete approaches such as SPH they are numerically more efficient but do not account for internal fracture flow dynamics, flow modes or other similar features.

We apply discrete models to get a better understanding of infiltration dynamics on outcrop and catchment scales. The connection between fracture (network) geometry/topology and the formation of preferential flow paths (flow instabilities) in fractured-porous media is still an open question and requires further research on numerical implementation, analytical modeling as well as (analogue) laboratory and field experiments.