Richards equation

Summary

The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited to specific initial and boundary conditions. Proof of the existence and uniqueness of solution was given only in 1983 by Alt and Luckhaus. The equation is based on Darcy-Buckingham law representing flow in porous media under variably saturated conditions, which is stated as :\vec{q}=-\mathbf{K}(\theta) (\nabla h + \nabla z), where :\vec{q} is the volumetric flux; :\theta is the volumetric water content; :h is the liquid pressure head, which is negative for unsaturated porous media; :\mathbf{K}(h) is the unsaturated hydraulic conductivity; :\nabla z is the geodetic head gradient, which is assumed as \nabla z = \left(\begin{smallmatrix} 0 \

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The modeling of unsaturated groundwater flow is affected by a high degree of uncertainty related to both measurement and model errors. Geophysical methods such as Electrical Resistivity Tomography (ERT) can provide useful indirect information on the hydrological processes occurring in the vadose zone. In this paper, we propose and test an iterated particle filter method to solve the coupled hydrogeophysical inverse problem. We focus on an infiltration test monitored by time-lapse ERT and modeled using Richards equation. The goal is to identify hydrological model parameters from ERT electrical potential measurements. Traditional uncoupled inversion relies on the solution of two sequential inverse problems, the first one applied to the ERT measurements, the second one to Richards equation. This approach does not ensure an accurate quantitative description of the physical state, typically violating mass balance. To avoid one of these two inversions and incorporate in the process more physical simulation constraints, we cast the problem within the framework of a SIR (Sequential Importance Resampling) data assimilation approach that uses a Richards equation solver to model the hydrological dynamics and a forward ERT simulator combined with Archie's law to serve as measurement model. ERT observations are then used to update the state of the system as well as to estimate the model parameters and their posterior distribution. The limitations of the traditional sequential Bayesian approach are investigated and an innovative iterative approach is proposed to estimate the model parameters with high accuracy. The numerical properties of the developed algorithm are verified on both homogeneous and heterogeneous synthetic test cases based on a real-world field experiment.

2015

Measuring the effect of structural connectivity on the water dynamics in heterogeneous porous media using speedy neutron tomography

The temporal and spatial distribution of water within a porous medium is affected by the medium's structure, i.e., the spatial arrangement of its constituents. To analyze structural effects on the fluid dynamics, we measured the 3D water content distribution in a heterogeneous sand column during two drainage-wetting cycles using neutron transmission tomography. The sample with a volume of 105 cm(3) contained 101 cubes of fine and 49 cubes of coarse sand with particles ranging from 0.01 to 0.05 and 0.03 to 0.09 cm, respectively. The pressure at the lower boundary was determined by the water reservoir positioned between 7 and 39 cm below the top of the column. The duration of one complete 3D scanning with a spatial resolution of 127 gm was 56 s. The signal to noise ratio of the measurements was low due to the short exposure time in the neutron beam, but it was possible to quantify the water content in the individual cubes and hence the effect of structure on macroscopic water distribution. Continuous structures of coarse sand drained faster than coarse sand without connection to the upper boundary. During the initial wetting phase, cubes of coarse sand material completely embedded in the fine material remained water unsaturated due to air entrapment. The effect of the coarse sand connectivity was analyzed in two-dimensional numerical simulations based on Richards equation. In contrast to the measurements, no effect of structure connectivity was found. The coarse sand cubes embedded within the fine matrix drain as quickly as the coarse sand cubes arranged in a continuous channel due to the model assumption of a continuous air phase. (C) 2008 Elsevier Ltd. All rights reserved.

2008

Scaling of capillary, gravity and viscous forces affecting flow morphology in unsaturated porous media

Dani Or

Interplay between capillary, gravity and viscous forces in unsaturated porous media gives rise to a range of complex flow phenomena affecting morphology, stability and dynamics of wetting and drainage fronts. Similar average phase contents may result in significantly different fluid distribution and patterns affecting macroscopic transport properties of the unsaturated medium. The formulation of general force balance within simplified pore spaces yields scaling relationships for motion of liquid elements in which gravitational force in excess of capillary pinning force scales linearly with viscous force. Displacement fluid front morphology is described using dimensionless force ratios expressed as Bond and Capillary numbers. The concise representations of a wide range of flow regimes with scaling relations, and predictive capabilities of front morphology based on dimensionless numbers lend support to certain generalizations. Considering available experimental data, we are able to define conditions for onset of unstable and intermittent flows leading to enhanced liquid and gas entrapment. These results provide a basis for delineation of a tentative value of Bo similar to 0.05 as an upper limit of applicability of the Richards equation (at pore to sample scales) and related continuum-based flow models. (C) 2007 Elsevier Ltd. All rights reserved.

2008