scholarly journals Richards' equation reconsidered

2021 ◽  
Author(s):  
Rostislav Vodák ◽  
Tomáš Fürst ◽  
Miloslav Šír ◽  
Jakub Kmec
Keyword(s):  
Porous Medium ◽  
Continuum Model ◽  
Block Size ◽  
Richards Equation ◽  
Porous Media Flow ◽  
Unsaturated Porous Media ◽  
Retention Curve ◽  
Hysteresis Operator ◽  
Continuum Modelling ◽  
Elasto Plasticity

Abstract Semi-continuum modelling of unsaturated porous media flow is based on representing the porous medium as a grid of non-infinitesimal blocks that retain the character of a porous medium. Semi-continuum model is able to physically correctly describe diffusion-like flow, finger-like flow, and the transition between them. This article presents the limit of the semi-continuum model as the block size goes to zero. In the limiting process, the retention curve of each block scales with the block size and in the limit becomes a hysteresis operator of the Prandtl-type used in elasto-plasticity models. Mathematical analysis showed that the limit of the semi-continuum model is a partial differential equation with a hysteresis operator of Prandl's type. This limit differs from the standard Richards' Equation (RE), which is not able to describe finger-like flow. Since the physics behind both RE and the semi-continuum model is almost the same, we suggest a way to reformulate the RE so that it retains the ability to describe finger-like flow. We conclude that RE should be reconsidered by means of appropriate modelling of the hysteresis and correct scaling of the retention curve.

scholarly journals A two dimensional semi-continuum model to explain wetting front instability in porous media

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jakub Kmec ◽  
Tomáš Fürst ◽  
Rostislav Vodák ◽  
Miloslav Šír
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Continuum Mechanics ◽  
Continuum Model ◽  
Preferential Flow ◽  
Finite Size ◽  
Porous Media Flow ◽  
Two Dimensional ◽  
Wetting Front ◽  
Saturation Overshoot

AbstractModelling fluid flow in an unsaturated porous medium is a complex problem with many practical applications. There is enough experimental and theoretical evidence that the standard continuum mechanics based modelling approach is unable to capture many important features of porous media flow. In this paper, a two-dimensional semi-continuum model is presented that combines ideas from continuum mechanics with invasion percolation models. The medium is divided into blocks of finite size that retain the nature of a porous medium. Each block is characterized by its porosity, permeability, and a retention curve. The saturation and pressure of the fluids are assumed to be uniform throughout each block. It is demonstrated that the resulting semi-continuum model is able to reproduce (1) gravity induced preferential flow with a spatially rich system of rivulets (fingers) characterized by saturation overshoot, (2) diffusion-like flow with a monotonic saturation profile, (3) the transition between the two. The model helps to explain the formation of the preferential pathways and their persistence and structure (the core and fringe of the fingers), the effect of the initial saturation of the matrix, and the saturation overshoot phenomenon.

scholarly journals Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170612 ◽  
Author(s):  
Malena I. Español ◽  
Dmitry Golovaty ◽  
J. Patrick Wilber
Keyword(s):  
Continuum Model ◽  
Domain Walls ◽  
Three Dimensional ◽  
Variational Model ◽  
Dimensional System ◽  
Two Dimensional ◽  
Starting Point ◽  
Continuum Modelling ◽  
Three Dimensional System ◽  
The Continuum

In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media

2021 ◽  
Author(s):  
Nicolae Suciu ◽  
Davide Illiano ◽  
Alexander Prechtel ◽  
Florin Radu
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Random Walk ◽  
Transport Processes ◽  
Richards Equation ◽  
Unsaturated Porous Media ◽  
Coupled Flow ◽  
Flow And Transport ◽  
Coupled Flow And Transport ◽  
Fully Coupled

<p>We present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the <em>L</em>-scheme developed in finite element/volume approaches. The resulting GRW <em>L</em>-schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards' equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.  For completeness, we also consider decoupled flow and transport model problems for saturated aquifers.</p>

scholarly journals A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

2019 ◽  
Vol 24 (1) ◽  
pp. 311-331 ◽  
Author(s):  
Prashant Kumar ◽  
Carmen Rodrigo ◽  
Francisco J. Gaspar ◽  
Cornelis W. Oosterlee
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Nonlinear Problems ◽  
Picard Iteration ◽  
Richards Equation ◽  
Porous Media Flow ◽  
Soil Parameters ◽  
Multilevel Monte Carlo ◽  
Highly Nonlinear ◽  
Non Gaussian

AbstractWe present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator.

scholarly journals Micropolar continuum modelling of bi-dimensional tetrachiral lattices

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130734 ◽  
Author(s):  
Y. Chen ◽  
X. N. Liu ◽  
G. K. Hu ◽  
Q. P. Sun ◽  
Q. S. Zheng
Keyword(s):  
Continuum Model ◽  
Rotational Symmetry ◽  
Tensor Decomposition ◽  
Additional Material ◽  
Micropolar Continuum ◽  
Material Constants ◽  
Continuum Modelling ◽  
Micropolar Model ◽  
Wave Properties ◽  
The Continuum

The in-plane behaviour of tetrachiral lattices should be characterized by bi-dimensional orthotropic material owing to the existence of two orthogonal axes of rotational symmetry. Moreover, the constitutive model must also represent the chirality inherent in the lattices. To this end, a bi-dimensional orthotropic chiral micropolar model is developed based on the theory of irreducible orthogonal tensor decomposition. The obtained constitutive tensors display a hierarchy structure depending on the symmetry of the underlying microstructure. Eight additional material constants, in addition to five for the hemitropic case, are introduced to characterize the anisotropy under Z 2 invariance. The developed continuum model is then applied to a tetrachiral lattice, and the material constants of the continuum model are analytically derived by a homogenization process. By comparing with numerical simulations for the discrete lattice, it is found that the proposed continuum model can correctly characterize the static and wave properties of the tetrachiral lattice.

scholarly journals A revised model of "Steady laminar natural convection of nanofluid under the impact of magnetic field on 2-D cavity with radiation" [AIP advances 9, 065008 (2019); https://doi.org/10.1063/1.5109192]

2020 ◽  
Vol 24 (1 Part A) ◽  
pp. 421-425
Author(s):  
Hossam Nabwey
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Natural Convection ◽  
Darcy Number ◽  
Porous Media Flow ◽  
Governing Equations ◽  
Laminar Natural Convection ◽  
The Impact ◽  
Published Paper ◽  
Steady Laminar

This discussion exhibits the major scientific errors on the recent published paper, entitled "Steady Laminar Natural Convection of Nanofluid Under the Impact of Magnetic Field on 2-D Cavity with Radiation" and their corrections indifferently. In Saleem et al. [1], the authors stated in both of abstract and problem assumptions that the non-Darcy model is used for the porous medium, while the porous terms are incompatible with this assumption. In addition, the authors used a non-inclined geometry in their investigation, but the governing equations are conflicting with this hypothesis. Further, the used range of the Darcy number is between 10?2-102 and this range is very large and did not represent the porous media flow. All of these observations make the mathematical formulations and the obtained results of Saleem et al. [1] are wrong. In the following sections, these scientific errors and their corrections will be presented minutely.

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