Multi-scale Smoothed Particle Hydrodynamics model for flow and transport in unsaturated fractured porous media
Palaeontology
Final Report Abstract
The main objective of the project was the development of an efficient multiscale Smoothed Particle Hydrodynamics (SPH) model to study infiltration dynamics in fractures and the adjacent porous matrix. This is achieved via discretization of the volume-effective Richards equation to govern porous medium flow, while the flow in fractures is realized by an SPH discretization of the Navier-Stokes equation. Flow from fractures into the porous matrix is modeled by an efficient particle removal algorithm and a virtual water redistribution formulation to enforce mass and momentum conservation. The developed model is to our knowledge the first SPH model that tightly couples the Navier-Stokes flow equations in a fracture with the continuum (Richards) model of flow in the adjacent porous matrix. Alternative approaches mostly treat both, fractures and porous matrix as continua and therefore are partially not able to describe the interaction dynamics that arise from the complex wetting patterns of fracture fluid on the porous fracture walls. Such approaches are based on the assumption that the system is well mixed within the elementary representative volume. Our approach does not rely on this assumption for flow in fracture (at the expense of computational resources) – yet, it can be used to: (1) test the limits of applicability of models that employ a continuum description of fractured porous media; (2) study the effect of boundary conditions on flow regimes in fractures; and (3) potentially improve continuum models. The model has been validated by (1) comparison to a finite element model (FEM) COMSOL for Richards-based flow dynamics in a partially saturated medium and (2) laboratory experiments to cover more complex cases of free-surface flow dynamics and imbibition into the porous matrix. For the laboratory experiments, Seeberger sandstone is used because of its well-known homogeneous pore space properties. The saturated hydraulic conductivity of the permeable matrix is estimated from a pore size and grain size distribution analysis. The saturation dynamics modelled with the developed PF-SPH model show good correlation with the COMSOL model and all types of laboratory experiments. Amongst others we employ the proposed model to study preferential flow dynamics for different infiltration rates. Here, flow in fracture is associated with the term “preferential flow", providing rapid water transmission, while flow within the adjacent porous matrix enables only slow and diffuse water transmission. Depending on the infiltration rate and water inlet location, two cases can be distinguished: (1) immediate preferential/fracture flow or (2) delayed preferential flow. In the latter case, water accumulates at the surface first (ponding), and then the fracture rapidly transmits water to the bottom system outlet. For the immediate fracture flow response, ponding only occurs once the fracture is fully saturated with water. In all cases, preferential flow is much more rapid than diffuse flow even under saturated porous medium conditions. Furthermore, infiltration dynamics in rough fractures adjacent to an impermeable or permeable matrix for different infiltration rates are studied as well. The simulation results show a significant lag in arrival times for small infiltration rates when a permeable porous matrix is employed, rather than an impermeable one. For higher infiltration rates, water rapidly flows through the fracture to the system outlet without any significant delay in arrival times even in the presence of the permeable matrix. The analysis of the amount of water stored in permeable fracture walls and in a fracture void space shows that for small infiltration rates, most of the injected water is retarded within the porous matrix. Flow velocity is higher for large infiltration rates, such that most of the water flows rapidly to the bottom of the fracture with very little influence of matrix imbibition processes. Finally, currently work in progress, we investigate fluid flow in horizontal initially dry fractures at low infiltration rate 𝑄 = 8 ∙ 10^-6 m^3 /s for different hydraulic conductivities of the adjacent porous matrix. Our results indicate that based on the hydraulic conductivity of the porous matrix the bulk flow can be divided into three regimes: (1) matrix saturation (porous medium flow dominates); (2) matrix saturation and fluid front propagation (porous medium and fracture flow happen simultaneously); (3) fluid propagation (fracture flow dominates). These preliminary studies are in alignment with classical theories yet also show interesting deviations when explicit geometries and fracture flow processes are considered.
Publications
- (2017): "Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow". Physical Review E, 96, p. 033115
Shigorina, E., Kordilla, J., Tartakovsky, A.
(See online at https://doi.org/10.1103/PhysRevE.96.033115) - (2019): "Investigation of Gravity-Driven Infiltration Instabilities in Smooth and Rough Fractures Using a Pairwise-Force Smoothed Particle Hydrodynamics Model ". Vadose Zone Journal, 18, p. 1-12
Shigorina, E., Kordilla, J., Tartakovsky, A.
(See online at https://doi.org/10.2136/vzj2018.08.0159) - (2019): "Preferential flow dynamics of the vadose zone of fractured and fracturedporous media: Development of a parallelized multi-scale Smoothed Particle Hydrodynamics Model", University of Göttingen, pp. 127, Dissertation
Shigorina, E.
- (2020): "Multiscale smoothed particle hydrodynamics model development for simulating preferential flow dynamics in fractured porous media". Water Resources Research, 2020WR027323R
Shigorina, E., Rüdiger, F., Tartakovsky, A., Sauter, M., Kordilla, J.
(See online at https://doi.org/10.1029/2020WR027323) - (2021): Numerical and analytical modeling of flow partitioning in partially saturated fracture networks. Water Resources Research, e2020WR028775
Kordilla, J., Tartakovsky, A., Dentz, M.
(See online at https://doi.org/10.1029/2020WR028775)
