Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium

Infiltration of water in dry porous media is subject to a powerful gravity-driven instability. Although the phenomenon of unstable infiltration is well known, its description using continuum mathematical models has posed a significant challenge for several decades. The classical model of water flow in the unsaturated flow, the Richards equation, is unable to reproduce the instability. Here, we present a computational study of a model of unsaturated flow in porous media that extends the Richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively. The extended model is based on a phase-field formulation and is fourth-order in space. The new model poses a set of challenges for numerical discretizations, such as resolution of evolving interfaces, stiffness in space and time, treatment of singularly perturbed equations, and discretization of higher-order spatial partial–differential operators. We develop a numerical algorithm based on Isogeometric Analysis, a generalization of the finite element method that permits the use of globally-smooth basis functions, leading to a simple and efficient discretization of higher-order spatial operators in variational form. We illustrate the accuracy, efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media. We simulate, for the first time, unstable gravity-driven infiltration in three dimensions, and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium. Our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux.United States. Dept. of Energy (Early Career Award Grant DE-SC0003907

Numerical Simulations of Gravity-Driven Fingering in Unsaturated Porous Media Using a Non-Equilibrium Model

This is a computational study of gravity-driven fingering instabilities in unsaturated porous media. The governing equations and corresponding numerical scheme are based on the work of Nieber et al. [Ch. 23 in Soil Water Repellency, eds. C. J. Ritsema and L. W. Dekker, Elsevier, 2003] in which non-monotonic saturation profiles are obtained by supplementing the Richards equation with a non-equilibrium capillary pressure-saturation relationship, as well as including hysteretic effects. The first part of the study takes an extensive look at the sensitivity of the finger solutions to certain key parameters in the model such as capillary shape parameter, initial saturation, and capillary relaxation coefficient. The second part is a comparison to published experimental results that demonstrates the ability of the model to capture realistic fingering behaviour

Numerical Simulation of Unstable Preferential Flow during Water Infiltration into Heterogeneous Dry Soil

Water infiltration and unsaturated flow through heterogeneous soil control the distribution of soil moisture in the vadose zone and the dynamics of groundwater recharge, providing the link between climate, biogeochemical soil processes and vegetation dynamics. Infiltration into dry soil is hydrodynamically unstable, leading to preferential flow through narrow wet regions (fingers). In this paper we use numerical simulation to study the interplay between fingering instabilities and soil heterogeneity during water infiltration. We consider soil with heterogeneous intrinsic permeability. Permeabilities are random, with point Gaussian statistics, and vary smoothly in space due to spatial correlation. The key research question is whether the presence of moderate or strong heterogeneity overwhelms the fingering instability, recovering the simple stable displacement patterns predicted by most simplified model of infiltration currently used in hydrological models from the Darcy to the basin scales. We perform detailed simulations of constant-rate infiltration into soils with isotropic and anisotropic intrinsic permeability fields. Our results demonstrate that soil heterogeneity does not suppress fingering instabilities, but it rather enhances its effect of preferential flow and channeling. Fingering patterns strongly depend on soil structure, in particular the correlation length and anisotropy of the permeability field. While the finger size and flow dynamics are only slightly controlled by correlation length in isotropic fields, layering leads to significant finger meandering and bulging, changing arrival times and wetting efficiencies. Fingering and soil heterogeneity need to be considered when upscaling the constitutive relationships of multiphase flow in porous media (relative permeability and water retention curve) from the finger to field and basin scales. While relative permeabilities remain unchanged upon upscaling for stable displacements, the inefficient wetting due to fingering leads to relative permeabilities at the field scale that are significantly different from those at the Darcy scale. These effective relative permeability functions also depend, although less strongly, on heterogeneity and soil structure

Dispersion enhancement and damping by buoyancy driven flows in 2D networks of capillaries

The influence of a small relative density difference on the displacement of two miscible liquids is studied experimentally in transparent 2D networks of micro channels. Both stable displacements in which the denser fluid enters at the bottom of the cell and displaces the lighter one and unstable displacements in which the lighter fluid is injected at the bottom and displaces the denser one are realized. Except at the lowest mean flow velocity U, the average $C(x,t)$ of the relative concentration satisfies a convection-dispersion equation. The dispersion coefficient is studied as function of the relative magnitude of fluid velocity and of the velocity of buoyancy driven fluid motion. A model is suggested and its applicability to previous results obtained in 3D media is discussed

Mass and Heat Flow through Snowpacks

Accurate estimation of snowmelt runoff is of primary importance in streamflow prediction for water management and flood forecasting in cold regions. Lateral flow, preferential flow pathways, and distinctive wetting and drying water retention curves in porous media have proven critical to improving soil water flow models; the most sophisticated physically based snowmelt models only account for 1D matrix flow and employ a single drying water retention curve for both drying and wetting snowpacks. Thus, there is an immediate need to develop snowmelt models that represent lateral and preferential flows, as well as full capillary hysteresis to examine the potential to improve snowmelt hydrological modelling. In this dissertation, the primary objective is to improve understanding and prediction of water flow through snow by investigating the formation of preferential flow paths and the coupling of heat and mass fluxes within snow. Of particular interest is the prediction of capillary pressure at macroscale, as it is of importance for simulating preferential flow in porous media. A novel 2D numerical model is developed that enables an improved understanding of energy and water flows within deep heterogeneous snowpacks on flat and sloping terrains. The numerical model simulates vertical and lateral water flow through snow matrix and preferential flow paths, and accounts for hysteresis in capillary pressure, internal energy fluxes, melt at the surface, and internal refreezing. Implementing a water entry pressure for initially dry snow was necessary for the formation of preferential flow paths. By coupling the simulation of preferential flow with heat transfer, ice layer formation was realistically simulated when water infiltrated an initially cold snowpack. Heat convection was added to the model and coupled to the energy balance at the snow surface; the transfer of heat by topography-driven airflow affected the estimated snow surface temperature by transporting thermal energy from the warm snow-soil interface to the upper snowpack. Comparisons of the model meltwater flow predictions against snowmelt field data revealed limitations in the current theories of water flow through snow, such as the use of a capillary entry pressure in the snow water retention curve that is limited to high-density snow. This suggested further concepts that would improve the representation of capillary pressure in snow models. This improved model, which considers a dynamic capillary pressure, gave better results than models based on previous theories when simulating capillary pressure overshoot. The research demonstrates how heterogeneous flow through snow can be modelled and how this research model furthers understanding of snowmelt flow processes and potential improvements in snowmelt-derived streamflow prediction

Fingering and strain localization in porous media during imbibition processes

Fingered infiltration of a wetting fluid through a porous network is a widely studied subject in the field of fluid mechanics. However, the effect of this heterogeneous percolation on the response of granular materials, in particular fine-grained soils, is a poorly investigated and badly understood topic which deserves deep analysis, considering, among others, possible applications in soil remediation and underground energy storage. This paper presents a first application of a new formulation of unsaturated poromechanics based on a phase field approach that allows to characterize on the one hand the occurrence of fingering hydraulic instabilities and on the other one to capture their effects on the irreversible, and possible unstable, deformation of the solid skeleton. The envisaged application concerns the behavior of fine-grained soils whose dilatant/contractant behavior is more and more attracting the interest of the scientific community both in the fields of experimental research and numerical modeling