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Numerical modeling of sharp-front water propagation in heterogeneous soils by the method of lines
Published in J.-L. Auriault, C. Geindreau, P. Royer, J.-F. Bloch, C. Boutin, J. Lewandowska, Poromechanics II, 2020
A. Szymkiewicz, J. Lewandowska
The unsaturated water flow in porous media is commonly described by a parabolic nonlinear partial differential equation known as Richards equation (Richards 1931) that in most cases can be solved only numerically. A number of numerical approaches has been developed, that vary in the choice of the form of the basic equation, approximation of the constitutive relations, spatial and temporal discretization and linearization method (Haverkamp et al. 1977, Celia et al. 1990 and others). Recently, intensive research has been carried out to elaborate robust and efficient solution to a class of non-linear sharp front propagation problems of great practical importance that are difficult to handle numerically. In result, a new approach was proposed i.e. DAE/MOL (Differential Algebraic Equations/ Method Of Lines) which is based on the application of higher order schemes of integration with respect to time (Tocci et al. 1997, Miller et al. 1998, Williams & Miller 1999). In this approach the basic equation is transformed into a system of ordinary differential equations by means of spatial discretization. Then, the integration in time can be done using one of the existing stiff solvers, like for example DASPK (Differential Algebraic Solver with Preconditioned Krylov method). DASPK is a variable-step size, variable-order differential algebraic equation code. It is believed that this approach (or similar ones) may provide efficient solutions for difficult nonlinear flow and transport problems (Tocci et al. 1997).
Spatial and material description of some processes in rigid and non-rigid saturated and unsaturated soils
Published in J.-F. Thimus, Y. Abousleiman, A.H.-D. Cheng, O. Coussy, E. Detournay, Poromechanics, 2020
functions of the liquid fraction θl, with the pc − θl relationship being time-invariant hysteretic (cf. Miller & Miller, 1956). For processes involving only monotonic changes in water content, using the unique pc − θl relationship gives: θlvl=−k(dpc/dθl)∇θl+kγlg=−D∇θl+kγlg, where D = kdpc / dθl is the diffusivity. Over the last fifty years a wide array of problems in soil physics and hydrology have been solved on the basis of this and related forms of the Richards equation. Generally these solutions have been obtained for specific soils defined by specific relationships between θl, pc and k (Raats, 1988, 1990, 1993). For one-dimensional, vertical downward flows, an important feature of some these solutions is a longtime limit in the form of a time- invariant traveling wave, arising as a so-called wetting front.
Complex Domain Flows
Published in Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin, Finite Analytic Method in Flows and Heat Transfer, 2020
Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin
Recently, the use of agricultural fertilizer, pesticides, and the treatment of wastewater have become great concerns to environmental and agricultural engineers. Chemical substances are frequently the source of soil and water pollutants. These pollutants are transported with water flow through the soil by several processes, including advection and diffusion. To estimate the magnitude of the hazard posed by these pollutants, it is important to investigate the processes that control the movement of water from the ground surface to the water table. The transport of water in unsaturated porous media plays an important role in environmental safety and agricultural applications. Accurate prediction of the transient water flow in unsaturated porous media is essential for the optimum management and control of groundwater contamination. Unfortunately, the Richards equation, which is the governing equation for water flow in unsaturated porous media, is highly nonlinear. Because of this, exact analytical solutions cannot be obtained, and numerical solutions must be sought.
Reliability-based optimization in climate-adaptive design of embedded footing
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Vahidreza Mahmoudabadi, Nadarajah Ravichandran
As is discussed in the previous section, in the first phase of the investigation, a parametric study is performed through the Monte Carlo simulation technique on three different footing cases. To this end, 10,000 random scenarios are analyzed using the site-specific input geotechnical and climatic parameters and their corresponding probability distribution. For each scenario, the Richards equation is solved with respect to the applied resultant infiltration and groundwater level as upper and lower boundary conditions, respectively. After computing the temporal and spatial soil water content and pressure head along with the subsurface soil profile, the arithmetic average of soil degree of saturation and matric suction is calculated for the footing influence zone. Finally, the serviceability and safety performance of each footing is estimated based on the footing ultimate bearing capacity and elastic settlement. This process repeats for all the scenarios at each of the study sites.
Application of physics-informed neural networks to inverse problems in unsaturated groundwater flow
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2022
Ivan Depina, Saket Jain, Sigurdur Mar Valsson, Hrvoje Gotovac
The Richards equation is a nonlinear PDE that captures the flow of water in unsaturated zone (Richards, 1931). One-dimensional form of the Richards equation is given by where θ is the volumetric soil water content(), t is the time (T), x is the vertical dimension (L), k is the hydraulic conductivity () and ψ is the pressure head (L). The WRC and HCF relationships can be specified by the van Genuchten model (Van Genuchten, 1980): where is the saturated hydraulic conductivity, Θ is the relative saturation, and are, respectively, the saturated and residual volumetric water contents, α (), n, and m are the model parameters, where . Equation (11a) specifies the HCF and Equation (11b) the WRC relationship.